Degree Graphs of Simple Groups

Donald L. WhiteDepartment of Mathematical Sciences

Kent State UniversityKent, Ohio 44242

E-mail: white@math.kent.edu

Version: October 16, 2006

ABSTRACT

Let G be a finite group and let cd(G) be the set of irreducible character degrees

of G. The degree graph ∆(G) is the graph whose set of vertices is the set of

primes that divide degrees in cd(G), with an edge between p and q if pq divides

a for some degree a ∈ cd(G). We compile here the graphs ∆(G) for all finite

simple groups G.

1. INTRODUCTION

The theory of characters is an important tool in the study of finitegroups. The irreducible complex characters of a finite group G encodemuch information about the structure of G. For example, using thesecharacters it is possible to determine the normal subgroups of G and there-fore whether G is simple. It is also possible to determine if G is abelian,nilpotent, or solvable.

Somewhat surprisingly, it is also possible to obtain some informationjust from the set of degrees of the characters, that is, their (integer) valueson the identity element of G. There has recently been much interest instudying the connections between the structure of a finite group G and thestructure of its set of character degrees. Of particular interest is the con-nection between the structure of G and common divisors among characterdegrees. A useful tool for studying this connection is the character degreegraph.

Let G be a finite group and let Irr(G) be the set of ordinary irreduciblecharacters of G. Denote the set of irreducible character degrees of G bycd(G) = χ(1)|χ ∈ Irr(G) and denote by ρ(G) the set of primes thatdivide degrees in cd(G). The character degree graph ∆(G) of G is thegraph whose set of vertices is ρ(G), with primes p, q in ρ(G) joined by anedge if pq divides a for some character degree a ∈ cd(G).

1

These graphs have been studied for some time, primarily for solvablegroups initially but more recently for nonsolvable groups as well. Some ofthe earliest results on character degree graphs were obtained by O. Manz,R. Staszewski, and W. Willems [13] on the number of connected compo-nents of the graph and by Manz, Willems, and T. R. Wolf [14] on thediameter of the graph for solvable groups. More recently, M. L. Lewisand the author have been able to classify nonsolvable groups G for which∆(G) is disconnected [10] and to bound the diameter of ∆(G) when G is anonsolvable group [11, 12]. See the article [9] by Lewis for an overview ofresults concerning ∆(G) and related graphs.

To each finite group G corresponds a unique collection of simple groupscalled the composition factors of G. In this sense, the finite simple groupsare the building blocks for finite groups. The composition factors of asolvable group are all cyclic groups of prime order. A nonsolvable group,however, has at least one nonabelian composition factor.

The basic approach for studying degree graphs and attacking othercharacter degree problems for nonsolvable groups is to first obtain as muchinformation as possible about the situation for the nonabelian finite simplegroups. The results are then extended to general nonsolvable groups usingthe information about their composition factors.

Of course, more complete results for the simple groups lead to betterresults for the nonsolvable groups. Using the partial results on degreegraphs of simple groups available at the time, Lewis and the author provedin [11] that, for a nonsolvable group G, the diameter of ∆(G) is at most 4.Later, using the complete results in [1], [18], [19], and [20], we were able tolower this bound to 3 in [12].

Our purpose in this survey is to describe the degree graphs for all non-abelian finite simple groups. By the classification of finite simple groups(see [7]), a nonabelian finite simple group must be an alternating groupAlt(n) with n > 5, a simple group of Lie type, or one of 26 sporadic simplegroups. The groups of Lie type may be further divided into the categoriesof groups of exceptional type and groups of classical type.

The irreducible characters of the 26 sporadic groups are available inthe Atlas [3]. The degree graphs for these groups are therefore knownimplicitly, but the graphs and proofs have not appeared in print previously.The situation is similar for the alternating groups Alt(n) for 5 6 n 6 14.The results of M. Barry and M. Ward in [1] determine ∆(Alt(n)) for n > 15.The graphs for the groups of Lie type were completely determined by theauthor in [18], [19], and [20].

We explicitly describe here the graphs for all nonabelian finite simplegroups and list the degrees used in the proofs. We give proofs in those casesfor which proofs have not previously appeared in print. For the other cases,we outline the proofs and give references to the complete published proofs.The sporadic groups are considered in §2, the alternating groups in §3,exceptional groups of Lie type in §4, and classical groups of Lie type in §5.

2

Finally, in §6, we summarize the results on connectedness and diameters ofthe degree graphs of the simple groups.

We say that a graph is a complete graph if there is an edge between everypair of vertices. Thus ∆(G) is a complete graph if whenever p and q are anydistinct primes dividing character degrees of G, there is some χ ∈ Irr(G)such that pq |χ(1). The graph ∆(G) tends to be a complete graph for simplegroups G. In fact, ∆(G) fails to be complete only for certain “small” simplegroups.

Moreover, many groups satisfy the stronger condition that there is acharacter χ ∈ Irr(G) such that every prime in ρ(G) divides χ(1). In thiscase, we say G has covering number 1. (The covering number is the smallestnumber of irreducible character degrees required to “cover” the primes inρ(G).) It is shown in [1] that Alt(n) has covering number 1 for all n > 15.

Notice that if ∆(G) is a complete graph but G does not have coveringnumber 1, then a proof that the graph is complete requires at least threecharacter degrees. In our proofs, particularly for the sporadic and smallalternating groups, we attempt to use the fewest possible degrees in eachcase. Of course, if ∆(G) is not complete, all degrees must be used in orderto determine the graph.

Our notation is mostly standard. Notation for the sporadic simplegroups is as in the Atlas [3], as are the labels for the characters of the smallalternating groups. We refer the reader to [2] for notation and results forthe simple groups of Lie type. We will use π(n) to denote the set of primedivisors of a positive integer n. If G is a nonabelian finite simple group, thenby the Ito-Michler Theorem (see [15, Remarks 13.13]), ρ(G) = π(|G|). Wewill also denote by Φk = Φk(q) the value of the kth cyclotomic polynomialevaluated at q.

2. SPORADIC SIMPLE GROUPS

We first describe the graph ∆(G) for a sporadic simple group G. Asnoted in the introduction, the explicit character tables of the sporadicgroups are known, so it is straightforward to determine the graphs. Thegraphs were described partially in [11, Lemma 2.3], but the graphs andproofs have not appeared in print elsewhere. It was shown in [11] that thesporadic simple group J1 is the only simple group whose degree graph hasdiameter greater than 2, and the diameter of ∆(J1) is 3.

3

Theorem 2.1. If G is a sporadic simple group other than M11, J1, orM23, then ∆(G) is a complete graph. The graphs of M11, J1, and M23 areas follows.

1. ∆(M11) is

3 5 PPP

2

11

2. ∆(J1) is

2 PPP

7

19

3

511

PPPPPP

3. ∆(M23) is

7

2

5

3

11

23HHH

HHH@@@

Proof. The character tables of the sporadic groups are found in theAtlas [3].

We have

cd(M11) = 1, 2 · 5, 11, 24, 22 · 11, 32 · 5, 5 · 11.

Hence ρ(M11) = 2, 3, 5, 11 and in ∆(G), the primes 2, 5, and 11 areall adjacent to each other, but 3 is adjacent only to 5, so the graph is asclaimed.

Next,

cd(J1) = 1, 23 · 7, 22 · 19, 7 · 11, 23 · 3 · 5, 7 · 19, 11 · 19

and ρ(J1) = 2, 3, 5, 7, 11, 19. The character degrees show that the primes3 and 5 are adjacent in ∆(J1), as are all of 7, 11, and 19, but neither of 3,5 is adjacent to any of 7, 11, 19. Also, 2 is adjacent to all of 3, 5, 7, and19, and the graph is as claimed.

We have

cd(M23) = 1, 2 · 11, 32 · 5, 2 · 5 · 23, 3 · 7 · 11, 11 · 23, 2 · 5 · 7 · 11,27 · 7, 2 · 32 · 5 · 11, 32 · 5 · 23, 23 · 11 · 23

4

and ρ(M23) = 2, 3, 5, 7, 11, 23. It is easily verified that the completegraph on 2, 3, 5, 7, 11 is a subgraph of ∆(M23) and that 23 is adjacent toeach of 2, 3, 5, and 11, but not to 7.

The orders of the remaining sporadic groups are listed in Table 1. It iseasily verified that ∆(G) is a complete graph by considering the characterdegrees in Table 2, Table 3, and Table 4. The groups listed in Table 4are those for which a single irreducible character degree is divisible by allprimes in π(|G|), that is, the sporadic groups with covering number 1.

3. ALTERNATING GROUPS

We next consider the simple alternating groups Alt(n) for n > 5. Thegraphs for these groups are complete except when n = 5, 6, or 8, and infact, Alt(n) has covering number 1 for all n > 15 by [1]. The exceptionsin this case are also special cases for the linear groups (§5.1), becauseAlt(5) ∼= PSL2(4) ∼= PSL2(5), Alt(6) ∼= PSL2(9), and Alt(8) ∼= PSL4(2).

Theorem 3.1. Let G = Alt(n), the alternating group of degree n, withn > 5. If n is not 5, 6, or 8, then the graph ∆(G) is complete. The graphsfor Alt(5), Alt(6), and Alt(8) are as follows.

1. ∆(Alt(5)) is

3r

5r2 r

2. ∆(Alt(6)) is

3r

5r2 r

3. ∆(Alt(8)) is

2 PPP

5

73PPP

Proof. For the graphs of Alt(5), Alt(6), and Alt(8), we have

cd(Alt(5)) = 1, 3, 22, 5,

cd(Alt(6)) = 1, 5, 23, 32, 2 · 5,and

cd(Alt(8)) = 1, 7, 2 · 7, 22 · 5, 3 · 7, 22 · 7, 5 · 7, 32 · 5, 23 · 7, 26, 2 · 5 · 7

5

by the Atlas [3] character tables. Therefore the graphs are as claimed. Wealso have

cd(Alt(7)) = 1, 2 · 3, 2 · 5, 2 · 7, 3 · 5, 3 · 7, 5 · 7,

so ∆(Alt(7)) is complete.The character degrees needed to prove that the graphs of Alt(n) are

complete for 9 6 n 6 14 are given in Table 5. The degrees for n 6 13 maybe found in the Atlas [3], and the degrees for Alt(14) were computed usingGAP [5].

It is shown by M. Barry and M. Ward in [1] that for n > 15, Alt(n) hasan irreducible character whose degree is divisible by all primes dividing theorder of Alt(n). Therefore ∆(Alt(n)) is complete for all n > 15.

4. EXCEPTIONAL GROUPS OF LIE TYPE

In this section, G is a simple group of exceptional Lie type. Thus Gwill be one of the following:

G2(q) for q 6= 2,

F4(q), E6(q), E7(q), E8(q),2E6(q

2), 3D4(q3) for all q,

2B2(q2) and 2F 4(q

2) for q2 = 22m+1, q2 6= 2,2F4(2)′,

2G2(q2) for q2 = 32m+1, q2 6= 3.

(See [2] or [3] for notation.) Note that certain values of q are excludedfor some types because the corresponding groups are not simple. Thegroup 2B2(2) is solvable. The derived groups of G2(2) and 2G2(3) aresimple but are isomorphic to the simple classical groups 2A2(3

2) and A1(8),respectively, which will be considered in §5. The derived group of 2F 4(2) issimple and does not appear elsewhere in the classification of finite simplegroups, so will be considered here.

Theorem 4.1. [18, Theorem 3.3] Let G ∼= 2B2(q2), where q2 = 22m+1

and m ≥ 1. The set of primes ρ(G) can be partitioned as

ρ(G) = 2 ∪ π(q2 − 1) ∪ π(q4 + 1).

The subgraph of ∆(G) on ρ(G)−2 is complete and 2 is adjacent in ∆(G)to precisely the primes in π(q2 − 1).

Proof. We have |G| = q4Φ1Φ2Φ8 and

cd(G) = 1, q4,1√2qΦ1Φ2, Φ8, Φ1Φ2Φ

′8, Φ1Φ2Φ

′′8,

where Φ′8 = q2 +

√2q + 1 and Φ′′

8 = q2 −√

2q + 1. (Thus Φ′8 and Φ′′

8 areintegers with Φ′

8Φ′′8 = Φ8.) See [18] for details.

6

Theorem 4.2. [18, Theorem 3.4] If G is a simple group of exceptionalLie type, other than type 2B2, then the graph ∆(G) is complete.

Proof. If G ∼= F4(2), then |G| = 224 ·36 ·52 ·72 ·13 ·17. By the Atlas [3],G has character degree χ46(1) = 1299480 = 23 · 3 · 5 · 72 · 13 · 17, so thegraph is complete.

If G ∼= 2F 4(2), then G′ ∼= 2F 4(2)′ is simple and |2F 4(2)′| = 211 ·33 ·52 ·13.By the Atlas [3], 2F 4(2)′ has character degrees χ6(1) = 78 = 2 · 3 · 13,χ7(1) = 300 = 22 · 3 · 52, and χ8(1) = 325 = 52 · 13, and these degrees showthat the graph is complete.

For the other cases, the group orders are given in Table 6 and thedegrees used to prove that the graphs are complete are given in Table 7and Table 8. See [2] for the notation for the character labels and see [18]for details of the proof.

5. CLASSICAL GROUPS OF LIE TYPE

The classical groups of Lie type are the classical simple linear, unitary,symplectic, and orthogonal groups. For the classical groups, it is generallyeasier to compute the degrees of the so-called adjoint group, which containsthe simple group of the given type as a normal subgroup. For the purposeof finding the character degree graphs, we will use the degrees for theadjoint groups along with the following well-known lemma (see [8, Corollary11.29]).

Lemma 5.1. If N E G with |G : N | = d, χ ∈ Irr(G), and µ is aconstituent of the restriction of χ to N , then χ(1)/d divides µ(1).

Various notations are used for these groups. In the terminology ofgroups of Lie type, the linear groups are of type Aℓ, the unitary groupsare of type 2Aℓ, the various orthogonal groups are of types Bℓ, Dℓ, and2Dℓ, and the symplectic groups are of type Cℓ. The table below gives theclassical matrix group notation and the notation used in the Atlas [3] forthese groups. Also listed are the adjoint group and the index of the simplegroup in the adjoint group for each type. The notation for the adjointgroups is as in [2].

Lie Simple Group AdjointType Classical Atlas Group Index

Aℓ PSLℓ+1(q) Lℓ+1(q) PGLℓ+1(q) (ℓ + 1, q − 1)

2Aℓ PSUℓ+1(q2) Uℓ+1(q) PUℓ+1(q) (ℓ + 1, q + 1)

Bℓ Ω2ℓ+1(q) O2ℓ+1(q) SO2ℓ+1(q) (2, q − 1)

Cℓ PSp2ℓ(q) S2ℓ(q) PCSp2ℓ(q) 2

Dℓ PΩ+2ℓ(q) O+

2ℓ(q) P(CO2ℓ(q)0) (4, qℓ − 1)

2Dℓ PΩ−2ℓ(q) O−

2ℓ(q) P(CO−2ℓ(q)

0) (4, qℓ + 1)

7

5.1. Type Aℓ: Linear Groups

In this section, G is the projective special linear group PSLℓ+1(q) oftype Aℓ, where q is a power of a prime p and ℓ > 1. If ℓ = 1, thenwe take q > 4, as PSL2(2) and PSL2(3) are not simple. Note also thatPSL2(5) ∼= PSL2(4) ∼= Alt(5).

Theorem 5.2. [19, Theorem 3.1] Let G ∼= PSL2(q), where q > 4 is apower of a prime p.

1. If q is even, then ∆(G) has three connected components, 2, π(q−1),and π(q + 1), and each component is a complete graph.

2. If q > 5 is odd, then ∆(G) has two connected components, p andπ ((q − 1)(q + 1)).

(a) The connected component π ((q − 1)(q + 1)) is a complete graphif and only if q − 1 or q + 1 is a power of 2.

(b) If neither of q−1 or q+1 is a power of 2, then π ((q − 1)(q + 1))can be partitioned as 2 ∪ M ∪ P , where M = π(q − 1) − 2and P = π(q + 1) − 2 are both nonempty sets. The subgraphof ∆(G) corresponding to each of the subsets M , P is complete,all primes are adjacent to 2, and no prime in M is adjacent toany prime in P .

Proof. If q = 2n, n > 2, then

cd(G) = 1, 2n − 1, 2n, 2n + 1.

If q = pn > 5 is odd, then

cd(G) = 1, q − 1, q, q + 1, (q + ǫ)/2,

where ǫ = (−1)(q−1)/2. See [19] for details.

We next describe the degree graph for PSL3(q). Note that PSL3(2) ∼=PSL2(7), so if q = 2, then the graph is described in Theorem 5.2. We willtherefore take q > 2.

8

Theorem 5.3. [19, Theorem 3.2] Let G ∼= PSL3(q), where q > 2 is apower of a prime p.

1. The graph ∆(G) is complete if and only if q is odd and q − 1 = 2i3j

for some i > 1, j > 0.

2. (a) If q = 4 then G ∼= PSL3(4) and ∆(G) is

2 5 PPP

3

7

(b) If q 6= 4, then ρ(G) = p ∪ π(

(q − 1)(q + 1)(q2 + q + 1))

. The

subgraph of ∆(G) corresponding to π(

(q − 1)(q + 1)(q2 + q + 1))

is complete and p is adjacent to precisely those primes dividingq + 1 or q2 + q + 1.

Proof. If q = 3, then G ∼= PSL3(3) and by the Atlas [3] character table,we have

cd(PSL3(3)) = 1, 22 · 3, 13, 24, 2 · 13, 33, 3 · 13.If q = 4, then G ∼= PSL3(4) and again by the character table in the Atlas [3],we have

cd(PSL3(4)) = 1, 22 · 5, 5 · 7, 32 · 5, 32 · 7, 26.If q > 4, then by the character table of G in either [16] or the CHEVIE

system [6], every character degree of G divides one of the degrees in thesubset

q3, q(q+1), (q−1)(q2+q+1), q(q2+q+1), (q+1)(q2+q+1), (q−1)2(q+1)

of cd(G). The theorem follows from these lists of degrees. See [19] fordetails.

We now consider the simple groups of type Aℓ for all ℓ > 3. The oneexceptional case is PSL4(2), which is isomorphic to the alternating groupAlt(8).

Theorem 5.4. [19, Theorem 3.3] Let G ∼= PSLℓ+1(q), where ℓ > 3 andq is a power of a prime p. The graph ∆(G) is complete unless ℓ = 3 andq = 2. If ℓ = 3 and q = 2, then G ∼= PSL4(2) ∼= Alt(8) and ∆(G) is

2 PPP

5

73PPP

Proof. The order of G ∼= PSLℓ+1(q) is given by

|G| =1

dqℓ(ℓ+1)/2(q2 − 1)(q3 − 1) · · · (qℓ − 1)(qℓ+1 − 1),

9

where d = (ℓ + 1, q − 1) = [PGLℓ+1(q) : PSLℓ+1(q)]. Therefore ρ(G) isprecisely the set of primes dividing q or some Φk for 1 6 k 6 ℓ + 1.

For ℓ > 4, the graph ∆(PSLℓ+1(q)) is seen to be complete using Lemma5.1 and the following degrees of PGLℓ+1(q):

χ(1,1,ℓ−1)(1) = q3 (qℓ−1 − 1)(qℓ − 1)

(q − 1)(q2 − 1)

χ1(1) = (q − 1)(q2 − 1)(q3 − 1) · · · (qℓ−1 − 1)(qℓ − 1)

χ2(1) = (q2 − 1)(q3 − 1) · · · (qℓ−1 − 1)(qℓ+1 − 1)

χ3(1) = q(q2 − 1)(q3 − 1) · · · (qℓ+1 − 1)

(q2 − 1)(qℓ−1 − 1).

For ℓ = 3, so G ∼= PSL4(q), the primes in ρ(G) are those primes dividingone of q, Φ1, Φ2, Φ3, or Φ4. In this case, PGL4(q) has the degrees

χ1(1) = Φ31Φ2Φ3

χ2(1) = Φ21Φ

22Φ4

χ3(1) = qΦ1Φ3Φ4.

For q = 3, these are sufficient to prove that the graph is complete. Forq > 3, PGL4(q) also has the degree χ(1) = qΦ2Φ3Φ4, and the graph isshown to be complete using this degree along with those listed above andLemma 5.1. Finally, if q = 2, then G ∼= PSL4(2) ∼= Alt(8), and

cd(G) = 1, 7, 2 · 7, 22 · 5, 3 · 7, 22 · 7, 5 · 7, 32 · 5, 23 · 7, 26, 2 · 5 · 7,

as noted in Theorem 3.1. The graph follows in this case. See [19] fordetails.

5.2. Type 2Aℓ: Unitary Groups

In this section, G is the projective special unitary group PSUℓ+1(q2)

of type 2Aℓ, where q is a power of a prime p. Note that this case is verysimilar to the case where G is of type Aℓ. Roughly speaking, replacing qwith −q in the results for type Aℓ yields the analogous results for type 2Aℓ.

Because PSU2(q2) ∼= PSL2(q), we take ℓ > 2. If ℓ = 2, then we take

q > 2, as PSU3(22) is not simple.

10

Theorem 5.5. [19, Theorem 3.4] Let G ∼= PSU3(q2), where q > 2 is a

power of a prime p.

1. The graph ∆(G) is complete if and only if q satisfies q + 1 = 2i3j forsome i > 0, j > 0.

2. If q > 2, then ρ(G) = p ∪ π(

(q − 1)(q + 1)(q2 − q + 1))

. The

subgraph of ∆(G) corresponding to π(

(q − 1)(q + 1)(q2 − q + 1))

iscomplete, and p is adjacent to precisely those primes dividing q − 1or q2 − q + 1.

Proof. Since q > 2, the character table of G in either [16] or the CHEVIE

system [6] shows that every character degree of G divides one of the degreesin the subset

q3, q(q−1), (q−1)(q2−q+1), q(q2−q+1), (q+1)(q2−q+1), (q−1)(q+1)2

of cd(G). The graph follows from this list of degrees. See [19] for details.

We next determine the graph ∆(G) for the simple groups of type 2Aℓ

for all ℓ > 3.

Theorem 5.6. [19, Theorem 3.5] If G ∼= PSUℓ+1(q2), where ℓ > 3 and

q is a power of a prime p, then the graph ∆(G) is complete.

Proof. The order of G ∼= PSUℓ+1(q2) is given by

|G| =1

dqℓ(ℓ+1)/2(q2 − 1)(q3 + 1) · · · (qℓ − (−1)ℓ)(qℓ+1 − (−1)ℓ+1),

where d = (ℓ + 1, q + 1) = [PUℓ+1(q2) : PSUℓ+1(q

2)]. Therefore ρ(G) isprecisely the set of primes dividing q or some qk − (−1)k for 2 6 k 6 ℓ + 1.

If (ℓ, q) is in (3, 2), (3, 3), (4, 2), then in each case there is an irreduciblecharacter whose degree is divisible by all primes in ρ(G). Using Atlas [3]notation for the characters, we have ρ(PSU4(2

2)) = 2, 3, 5 and χ11(1) =30 = 2 · 3 · 5, ρ(PSU4(3

2)) = 2, 3, 5, 7 and χ8(1) = 210 = 2 · 3 · 5 · 7, andρ(PSU5(2

2)) = 2, 3, 5, 11 and χ26(1) = 330 = 2 · 3 · 5 · 11. Hence ∆(G) iscomplete in these cases.

If (ℓ, q) 6∈ (3, 2), (3, 3), (4, 2), then PUℓ+1(q2) has the character de-

grees

χ(1,1,ℓ−1)(1) = q3 (qℓ−1 − (−1)ℓ−1)(qℓ − (−1)ℓ)

(q + 1)(q2 − 1)

χ1(1) = (q + 1)(q2 − 1)(q3 + 1) · · · (qℓ−1 − (−1)ℓ−1)(qℓ − (−1)ℓ)

χ2(1) = (q2 − 1)(q3 + 1) · · · (qℓ−1 − (−1)ℓ−1)(qℓ+1 − (−1)ℓ+1)

χ3(1) = q(q2 − 1)(q3 + 1) · · · (qℓ+1 − (−1)ℓ+1)

(q2 − 1)(qℓ−1 − (−1)ℓ−1).

11

For ℓ > 4, these degrees, along with Lemma 5.1, are sufficient to prove thegraph is complete.

If ℓ = 3, so G ∼= PSU4(q2), then the primes in ρ(G) are those primes

dividing one of q, Φ1, Φ2, Φ4, or Φ6. In this case, since q > 3, PU4(q2) has

degrees

χ1(1) = Φ1Φ32Φ6

χ2(1) = Φ21Φ

22Φ4

χ3(1) = qΦ2Φ4Φ6

χ(1) = qΦ21Φ4.

The graph is shown to be complete for ℓ = 3 and q > 3 using these degreesand Lemma 5.1. See [19] for details.

5.3. Type Bℓ: Odd Dimensional Orthogonal Groups

In this section, G is the simple odd dimensional orthogonal groupΩ2ℓ+1(q) (or O2ℓ+1(q) in Atlas [3] notation) of type Bℓ, where q is a powerof a prime p. Since B1(q) ∼= A1(q), we assume ℓ > 2, and since B2(2) isisomorphic to the symplectic group Sp4(2), which is not simple, we assumeq > 2 when ℓ = 2.

Theorem 5.7. [20, Theorem 1.1] If G ∼= Ω2ℓ+1(q) is a simple group oftype Bℓ, where ℓ > 2 and q is a power of a prime p, then the graph ∆(G)is complete.

Proof. The order of G is given by

|G| =1

dqℓ2(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1),

where d = (2, q − 1) = [SO2ℓ+1(q) : Ω2ℓ+1(q)]. It follows that ρ(G) consistsof p and the primes dividing Φj or Φ2j for j = 1, . . . , ℓ.

If ℓ > 3, then SO2ℓ+1(q) has character degrees

χα(1) =1

2q4 (qℓ−2 − 1)(qℓ−1 − 1)(qℓ−1 + 1)(qℓ + 1)

(q2 − 1)2

χc(1) =(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1)

qℓ + 1

= (q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(qℓ − 1)

χ1(1) = q(q4 − 1)(q6 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1)

qℓ−1 + 1

(see [20, Lemma 2.2]). These degrees, along with Lemma 5.1, are sufficientto show ∆(G) is complete for ℓ > 3.

12

Now let ℓ = 2 so G ∼= B2(q) ∼= C2(q), hence G ∼= PSp4(q). As notedabove, G is not simple if q = 2. If q = 3, then |G| = 26 · 34 · 5 and, by theAtlas [3] character table, G has the character degree χ11(1) = 30 = 2 · 3 · 5,so the graph is complete. Hence we may assume q > 3. The order of G is

|G| = |PSp4(q)| =1

(2,Φ1)q4Φ2

1Φ22Φ4,

and the character tables in [17] and [4] show that for q > 3, G has thecharacter degrees

χ1(1) = Φ21Φ

22

χ2(1) = qΦ1Φ4

χ3(1) = qΦ2Φ4.

These degrees show that the graph ∆(G) is complete for ℓ = 2. See [20]for details.

5.4. Type Cℓ: Symplectic Groups

In this section, G is the simple symplectic group PSp2ℓ(q) of type Cℓ,where q is a power of a prime p. Since C2(q) ∼= B2(q) for all q, we assumeℓ > 3.

Theorem 5.8. [20, Theorem 1.1] If G ∼= PSp2ℓ(q) is a simple group oftype Cℓ, where ℓ > 3 and q is a power of a prime p, then the graph ∆(G)is complete.

Proof. Since Cℓ(q) ∼= Bℓ(q) for all ℓ if q is even, by Theorem 5.7 we mayassume q is odd. Since q is odd, the order of G ∼= PSp2ℓ(q) is

|G| =1

2qℓ2(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1).

It follows that ρ(G) consists of p and the primes dividing Φj or Φ2j forj = 1, . . . , ℓ.

If ℓ > 4, then the projective conformal symplectic group PCSp2ℓ(q) hascharacter degrees

χα(1) = q3 (q2(ℓ−2) − 1)(q2ℓ − 1)

(q2 − 1)2

χc(1) =(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1)

qℓ + 1

= (q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(qℓ − 1)

χ1(1) = q2 (q2 + 1)(q6 − 1)(q8 − 1) · · · (q2(ℓ−1) − 1)(q2ℓ − 1)

qℓ−2 + 1

13

(see [20, Lemma 2.3]), and we have [PCSp2ℓ(q) : PSp2ℓ(q)] = 2. UsingLemma 5.1, these degrees are sufficient to show that ∆(G) is complete forℓ > 4.

Now let ℓ = 3 so that G ∼= PSp6(q). The order of G is

|G| = |PSp6(q)| =1

2q9Φ3

1Φ32Φ3Φ4Φ6.

Since q is odd, the character table of the conformal symplectic groupCSp6(q) is available in the CHEVIE system [6]. This table shows thatPCSp6(q) has a character of degree

χ82(1, 1, q − 2)(1) = qΦ1Φ2Φ3Φ4Φ6,

and since [PCSp6(q) : PSp6(q)] = 2, this degree, along with Lemma 5.1,shows that ∆(G) is complete. See [20] for details.

5.5. Types Dℓ and 2Dℓ: Even Dimensional Orthogonal Groups

In this section, G is a simple even dimensional orthogonal group, so iseither PΩ+

2ℓ(q) of type Dℓ or PΩ−2ℓ(q) of type 2Dℓ (that is, O+

2ℓ(q) or O−2ℓ(q),

respectively, in Atlas [3] notation), where q is a power of a prime p. SinceDℓ

∼= Aℓ and 2Dℓ∼= 2Aℓ if ℓ is less than 4, we assume ℓ > 4.

Theorem 5.9. [20, Theorem 1.1] If G ∼= PΩ+2ℓ(q) is a simple group of

type Dℓ, where ℓ > 4 and q is a power of a prime p, then the graph ∆(G)is complete.

Proof. The order of G is given by

|G| =1

dqℓ(ℓ−1)(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(qℓ − 1),

where d = (4, qℓ − 1) = [P(CO2ℓ(q)0) : PΩ+

2ℓ(q)]. (See [2, §1.19] for nota-tion.) It follows that ρ(G) consists of p, the primes dividing Φj or Φ2j forj = 1, . . . , ℓ − 1, and the primes dividing Φℓ.

If ℓ ≥ 6, then P(CO2ℓ(q)0) has character degrees

χα(1) = q6 (qℓ−4 + 1)(q2(ℓ−3) − 1)(q2(ℓ−1) − 1)(qℓ − 1)

(q2 − 1)2(q4 − 1)

χc(1) =(q2 − 1)(q4 − 1) · · · (q2(ℓ−2) − 1)(q2(ℓ−1) − 1)(qℓ − 1)

(q + 1)(qℓ−1 + 1)

χ1(1) = q2 (q2 + 1)(q6 − 1) · · · (q2(ℓ−2) − 1)(q2(ℓ−1) − 1)(qℓ − 1)

(q + 1)(qℓ−3 + 1)

(see [20, Lemma 2.4]). Using Lemma 5.1, these degrees are sufficient toshow ∆(G) is complete when ℓ > 6.

14

If ℓ = 5, then

|G| =1

dq20Φ5

1Φ42Φ3Φ

24Φ5Φ6Φ8

and P(CO10(q)0) has character degree

χ1(1) = q2Φ31Φ2Φ3Φ4Φ5Φ6Φ8.

Since [P(CO10(q)0) : G] = (4, q5 − 1) = d, it follows using Lemma 5.1 that

∆(G) is complete for ℓ = 5.Finally, let ℓ = 4. If q = 2, then |G| = 212 · 35 · 52 · 7 and by the

character table of G in the Atlas [3], G has an irreducible character of degreeχ11(1) = 210 = 2 ·3 ·5 ·7. If q = 3, then |G| = 212 ·312 ·52 ·7 ·13 and again bythe Atlas character table, G has the degree χ17(1) = 5460 = 22 ·3 ·5 ·7 ·13.Hence ∆(G) is a complete graph if q = 2 or q = 3.

If ℓ = 4 and q > 3, then

|G| =1

dq12Φ4

1Φ42Φ3Φ

24Φ6

and P(CO8(q)0) has character degrees

χ1(1) = q2Φ21Φ3Φ

24Φ6

χc(1) = Φ41Φ

22Φ3Φ

24

χβ(1) =1

2q3Φ4

2Φ6.

Since [P(CO8(q)0) : G] = (4, q4 − 1) = d, these degrees, along with

Lemma 5.1, show that ∆(G) is complete in this case as well. See [20]for details.

Theorem 5.10. [20, Theorem 1.1] If G ∼= PΩ−2ℓ(q) is a simple group of

type 2Dℓ, where ℓ > 4 and q is a power of a prime p, then the graph ∆(G)is complete.

Proof. The order of G is given by

|G| =1

dqℓ(ℓ−1)(q2 − 1)(q4 − 1) · · · (q2(ℓ−1) − 1)(qℓ + 1),

where d = (4, qℓ + 1) = [P(CO−2ℓ(q)

0) : PΩ−2ℓ(q)]. (See [2, §1.19] for nota-

tion.) It follows that ρ(G) consists of p, the primes dividing Φj or Φ2j forj = 1, . . . , ℓ − 1, and the primes dividing Φ2ℓ.

If ℓ > 5, then P(CO−2ℓ(q)

0) has character degrees

χα(1) =1

2q3 (qℓ−3 − 1)(qℓ−2 + 1)(qℓ−1 − 1)(qℓ + 1)

(q2 + 1)(q − 1)2

χc(1) = (q2 − 1)(q4 − 1) · · · (q2(ℓ−2) − 1)(q2(ℓ−1) − 1)

χ1(1) = q2 (q2 + 1)(q6 − 1) · · · (q2(ℓ−2) − 1)(q2(ℓ−1) − 1)(qℓ + 1)

qℓ−2 + 1

15

(see [20, Lemma 2.5]). Using Lemma 5.1, these degrees are sufficient toshow ∆(G) is complete when ℓ > 5.

If ℓ = 4 then

|G| =1

dq12Φ3

1Φ32Φ3Φ4Φ6Φ8

and P(CO−8 (q)0) has character degrees

χc(1) = Φ31Φ

32Φ3Φ4Φ6,

χ1(1) = q2Φ1Φ2Φ3Φ6Φ8,

χ2(1) = q3Φ1Φ3Φ4Φ8.

Since [P(CO−8 (q)0) : G] = (4, q4 + 1) = d, these degrees, along with

Lemma 5.1, show that ∆(G) is complete for ℓ = 4. See [20] for details.

6. DIAMETERS OF DEGREE GRAPHS

Combining the results above, we have the following theorem summa-rizing results on connectedness and diameters of degree graphs of simplegroups. This result is used in [12] to improve the bound on the diameterof the degree graph for nonsolvable groups found in [11].

Theorem 6.1. [20, Corollary 1.2] Let G be a finite simple group. Thegraph ∆(G) is disconnected if and only if G ∼= PSL2(q) for some primepower q. If ∆(G) is connected, then the diameter of ∆(G) is at most 3 and∆(G) is a complete graph except in the following cases:

1. The diameter of ∆(G) is 3 if and only if G ∼= J1.

2. The diameter of ∆(G) is 2 if and only if G is isomorphic to one of

(a) the sporadic Mathieu group M11 or M23,

(b) the alternating group A8,

(c) the Suzuki group 2B2(q2), where q2 = 22m+1 and m > 1,

(d) the linear group PSL3(q), where q > 2 is even or q is odd andq − 1 is divisible by a prime other than 2 or 3, or

(e) the unitary group PSU3(q2), where q > 2 and q + 1 is divisible

by a prime other than 2 or 3.

16

REFERENCES

[1] M. J. J. Barry and M. B. Ward, On a conjecture of Alvis, J.Algebra 294 (2005), 136–155.

[2] R. W. Carter, “Finite Groups of Lie Type: Conjugacy Classes andComplex Characters,” Wiley, New York, 1985.

[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and

R. A. Wilson, “Atlas of Finite Groups,” Oxford University Press,London, 1984.

[4] H. Enomoto, The characters of the finite symplectic group Sp(4, q),q = 2f , Osaka J. Math. 9 (1972), 75–94.

[5] The GAP Group, GAP – Groups, Algorithms, and Programming,Version 4.4, 2006, (http://www.gap-system.org).

[6] M. Geck, G. Hiss, F. Lubeck, G. Malle, and G. Pfeiffer,CHEVIE — A system for computing and processing generic charactertables, AAECC 7 (1996), 175–210.

[7] D. Gorenstein, R. Lyons, and R. M. Solomon, “The Classification ofthe Finite Simple Groups,” Surveys and Monographs 40, #1, Amer.Math. Soc., Providence, RI (1994).

[8] I. M. Isaacs, “Character Theory of Finite Groups,” Academic Press,San Diego, California, 1976.

[9] M. L. Lewis, An overview of graphs associated with character degreesand conjugacy class sizes in finite groups, Rocky Mountain J. Math.,in press.

[10] M. L. Lewis and D. L. White, Connectedness of degree graphsof nonsolvable groups, J. Algebra 266 (2003), 51-76; Corrigendum to“Connectedness of degree graphs of nonsolvable groups” [J. Algebra266 (2003) 51–76], J. Algebra 290 (2005), 594–598.

[11] M. L. Lewis and D. L. White, Diameters of degree graphs of non-solvable groups, J. Algebra 283 (2005), 80-92.

[12] M. L. Lewis and D. L. White, Diameters of degree graphs of non-solvable groups II, submitted.

[13] O. Manz, R. Staszewski, and W. Willems, On the number of compo-nents of a graph related to character degrees, Proc. AMS 103 (1988),31-37.

[14] O. Manz, W. Willems, and T. R. Wolf, The diameter of the characterdegree graph, J. reine angew. Math. 402 (1989), 181-198.

17

[15] O. Manz and T. R. Wolf, “Representations of Solvable Groups,”Cambridge University Press, Cambridge, 1993.

[16] W. A. Simpson and J. S. Frame, The character tables for SL(3, q),SU(3, q2), PSL(3, q), PSU(3, q2), Can. J. Math. 25 (1973), 486–494.

[17] B. Srinivasan, The characters of the finite symplectic group Sp(4, q),Trans. Amer. Math. Soc. 131 (1968), 488–525.

[18] D. L. White, Degree graphs of simple groups of exceptional Lie type,Comm. Algebra 32 (9) (2004), 3641-3649.

[19] D. L. White, Degree graphs of simple linear and unitary groups,Comm. Algebra 34 (8) (2006), 2907–2921.

[20] D. L. White, Degree graphs of simple orthogonal and symplecticgroups, J. Algebra, in press.

18

APPENDIX: TABLES OF ORDERS AND DEGREES

TABLE 1Orders of Sporadic Groups

Group Order

M12 26 · 33 · 5 · 11

M22 27 · 32 · 5 · 7 · 11

J2 27 · 33 · 52 · 7HS 29 · 32 · 53 · 7 · 11

J3 27 · 35 · 5 · 17 · 19

M24 210 · 33 · 5 · 7 · 11 · 23

M cL 27 · 36 · 53 · 7 · 11

He 210 · 33 · 52 · 73 · 17

Ru 214 · 33 · 53 · 7 · 13 · 29

Suz 213 · 37 · 52 · 7 · 11 · 13

O′N 29 · 34 · 5 · 73 · 11 · 19 · 31

Co3 210 · 37 · 53 · 7 · 11 · 23

Co2 218 · 36 · 53 · 7 · 11 · 23

Fi22 217 · 39 · 52 · 7 · 11 · 13

HN 214 · 36 · 56 · 7 · 11 · 19

Ly 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67

Th 215 · 310 · 53 · 72 · 13 · 19 · 31

Fi23 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23

Co1 221 · 39 · 54 · 72 · 11 · 13 · 23

J4 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43

Fi′24 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29

B 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47

M 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

19

TABLE 2Degrees of Sporadic Groups

Group Char Degree

M12 χ8 55 = 5 · 11χ11 66 = 2 · 3 · 11χ13 120 = 23 · 3 · 5

M22 χ5 55 = 5 · 11χ6 99 = 32 · 11χ7 154 = 2 · 7 · 11χ8 210 = 2 · 3 · 5 · 7

J2 χ7 63 = 32 · 7χ8 70 = 2 · 5 · 7χ10 90 = 2 · 32 · 5

HS χ4 154 = 2 · 7 · 11χ13 825 = 3 · 52 · 11χ22 2520 = 23 · 32 · 5 · 7

J3 χ9 816 = 24 · 3 · 17χ10 1140 = 22 · 3 · 5 · 19χ13 1615 = 5 · 17 · 19

M24 χ10 770 = 2 · 5 · 7 · 11χ14 1035 = 32 · 5 · 23χ21 3312 = 24 · 32 · 23χ23 5313 = 3 · 7 · 11 · 23

M cL χ3 231 = 3 · 7 · 11χ5 770 = 2 · 5 · 7 · 11χ12 4500 = 22 · 32 · 53

20

TABLE 3Degrees of Sporadic Groups, Continued

Group Char Degree

He χ12 1920 = 27 · 3 · 5χ17 7497 = 32 · 72 · 17χ25 11900 = 22 · 52 · 7 · 17

Ru χ11 27000 = 23 · 33 · 53

χ21 52780 = 22 · 5 · 7 · 13 · 29χ24 71253 = 33 · 7 · 13 · 29

Suz χ3 364 = 22 · 7 · 13χ9 5940 = 22 · 33 · 5 · 11χ13 15015 = 3 · 5 · 7 · 11 · 13

O′N χ12 58311 = 32 · 11 · 19 · 31χ25 175770 = 2 · 34 · 5 · 7 · 31χ29 234080 = 25 · 5 · 7 · 11 · 19

Co3 χ13 5544 = 23 · 32 · 7 · 11χ22 31625 = 53 · 11 · 23χ27 57960 = 23 · 32 · 5 · 7 · 23

Ly χ9 381766 = 2 · 7 · 11 · 37 · 67χ10 1152735 = 3 · 5 · 31 · 37 · 67χ33 28787220 = 22 · 32 · 5 · 7 · 11 · 31 · 67

J4 χ19 35411145 = 3 · 5 · 7 · 11 · 23 · 31 · 43χ25 300364890 = 2 · 3 · 5 · 7 · 29 · 31 · 37 · 43χ42 1182518964 = 22 · 32 · 113 · 23 · 29 · 37

21

TABLE 4More Degrees of Sporadic Groups

Grp Char Degree

Co2 χ28 212520 = 23 · 3 · 5 · 7 · 11 · 23

Fi22 χ24 150150 = 2 · 3 · 52 · 7 · 11 · 13

HN χ29 1053360 = 24 · 32 · 5 · 7 · 11 · 19

Th χ39 40199250 = 2 · 3 · 53 · 7 · 13 · 19 · 31

Fi23 χ43 35225190 = 2 · 32 · 5 · 7 · 11 · 13 · 17 · 23

Co1 χ36 9669660 = 22 · 3 · 5 · 72 · 11 · 13 · 23

Fi′24 χ34 7150713570 = 2 · 32 · 5 · 72 · 11 · 13 · 17 · 23 · 29

B χ65 40955835260340 = 22 · 33 · 5 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47

M χ188 198203900044423845494482560 =27 · 32 · 5 · 74 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

22

TABLE 5Orders and Degrees of Alternating Groups

Group Order Char Degree

Alt(9) 26 · 34 · 5 · 7 χ8 35 = 5 · 7χ9 42 = 2 · 3 · 7χ14 120 = 23 · 3 · 5

Alt(10) 27 · 34 · 52 · 7 χ11 210 = 2 · 3 · 5 · 7

Alt(11) 27 · 34 · 52 · 7 · 11 χ31 2310 = 2 · 3 · 5 · 7 · 11

Alt(12) 29 · 35 · 52 · 7 · 11 χ10 320 = 26 · 5χ12 462 = 2 · 3 · 7 · 11χ19 1155 = 3 · 5 · 7 · 11

Alt(13) 29 · 35 · 52 · 7 · 11 · 13 χ6 220 = 22 · 5 · 11χ36 6006 = 2 · 3 · 7 · 11 · 13χ52 15015 = 3 · 5 · 7 · 11 · 13

Alt(14) 210 · 35 · 52 · 72 · 11 · 13 χ10 715 = 5 · 11 · 13χ24 6006 = 2 · 3 · 7 · 11 · 13χ36 13650 = 2 · 3 · 52 · 7 · 13

23

TABLE 6Orders of Exceptional Groups

Group Order

G2(q) q6Φ21Φ

22Φ3Φ6

F4(q) q24Φ41Φ

42Φ

23Φ

24Φ

26Φ8Φ12

E6(q) q36Φ61Φ

42Φ

33Φ

24Φ5Φ

26Φ8Φ9Φ12

E7(q) q63Φ71Φ

72Φ

33Φ

24Φ5Φ

36Φ7Φ8Φ9Φ10Φ12Φ14Φ18

E8(q) q120Φ81Φ

82Φ

43Φ

44Φ

25Φ

46Φ7Φ

28Φ9Φ

210Φ

212Φ14Φ15Φ18Φ20Φ24Φ30

2B2(q2) q4Φ1Φ2Φ8

3D4(q3) q12Φ2

1Φ22Φ

23Φ

26Φ12

2G2(q2) q6Φ1Φ2Φ4Φ12

2F4(q2) q24Φ2

1Φ22Φ

24Φ

28Φ12Φ24

2E6(q2) q36Φ4

1Φ62Φ

23Φ

24Φ

36Φ8Φ10Φ12Φ18

24

TABLE 7Character Degrees of Exceptional Groups

Group Label Degree

G2(q) φ2,212qΦ2

2Φ6

G2[−1] 12qΦ2

1Φ3

Xa, χ10, χ6 Φ1Φ2Φ3Φ6

F4(q)q 6= 2

φ′8,3 q3Φ2

4Φ8Φ12

B2, r14q4Φ2

1Φ22Φ

23Φ

26Φ8

χs Φ41Φ

22Φ

23Φ

24Φ6Φ8Φ12

E6(q) φ64,4 q4Φ32Φ

24Φ

26Φ8Φ12

D4, 112q3Φ4

1Φ23Φ5Φ9

χs1Φ6

1Φ42Φ

33Φ

24Φ5Φ

26Φ8Φ12

χs2Φ6

1Φ42Φ

23Φ

24Φ5Φ

26Φ8Φ9

φ60,5 q5Φ4Φ5Φ8Φ9Φ12

E7(q) φ512,1212q11Φ7

2Φ24Φ

36Φ8Φ10Φ12Φ14Φ18

E7[ξ]12q11Φ7

1Φ33Φ

24Φ5Φ7Φ8Φ9Φ12

χs Φ71Φ

62Φ

33Φ

24Φ5Φ

36Φ7Φ8Φ9Φ10Φ12Φ14

D4, σ212q10Φ4

1Φ23Φ5Φ7Φ8Φ9Φ14Φ18

φ512,1112q11Φ7

2Φ24Φ

36Φ8Φ10Φ12Φ14Φ18

E8(q) φ4096,1212q11Φ7

2Φ44Φ

46Φ

28Φ

210Φ

212Φ14Φ18Φ20Φ24Φ30

E7[ξ], 112q11Φ7

1Φ43Φ

44Φ

25Φ7Φ

28Φ9Φ

212Φ15Φ20Φ24

φ1344,1914q16Φ4

2Φ24Φ

26Φ7Φ

28Φ9Φ

210Φ12Φ14Φ15Φ18Φ20Φ24Φ30

D4, φ16,514q16Φ4

1Φ42Φ

23Φ

25Φ

26Φ7Φ

28Φ9Φ

210Φ14Φ15Φ18Φ24Φ30

D4, φ′2,4

12q4Φ4

1Φ23Φ

24Φ

25Φ7Φ9Φ10Φ12Φ15Φ20Φ30

25

TABLE 8Character Degrees of Twisted Exceptional Groups

Group Label Degree

3D4(q3) 3D4[−1] 1

2q3Φ21Φ

23

φ2,112q3Φ2

2Φ26

φ′1,3 qΦ12

χ8 Φ1Φ2Φ3Φ26Φ12

2G2(q2) cuspidal 1√

3qΦ1Φ2Φ4

cuspidal 12√

3qΦ1Φ2Φ

′12

cuspidal 12√

3qΦ1Φ2Φ

′′12

ξ2 Φ12

η−i Φ1Φ2Φ4Φ

′′12

η+i Φ1Φ2Φ4Φ

′12

2F4(q2) 2B2[a], 1 1√

2qΦ1Φ2Φ

24Φ12

ρ212q4Φ2

8Φ24

χs Φ21Φ

22Φ

24Φ8Φ12Φ24

2E6(q2) 2A5, 1 q4Φ3

1Φ23Φ

24Φ8Φ12

φ′8,3

12q3Φ4

2Φ26Φ10Φ18

χs1Φ4

1Φ62Φ

23Φ

24Φ

36Φ8Φ10Φ12

χs2Φ4

1Φ62Φ

23Φ

24Φ

26Φ8Φ10Φ18

φ′4,7 q5Φ4Φ8Φ10Φ12Φ18

26