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1988

Bhaskar Rao designs over small groupsWilliam D. Palmer

Jennifer SeberryUniversity of Wollongong, jennie@uow.edu.au

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Publication DetailsPalmer, WD and Seberry, J, Bhaskar Rao designs over small groups, Ars Combinatoria, 26A, 1988, 125-148.

Bhaskar Rao designs over small groups

AbstractWe show that for each of the groups S3, D4, Q4, Z4 x Z2 and D6 the necessary conditions are sufficient for theexistence of a generalized Bhaskar Rao design. That is, we show that: (i) a GBRD (v, 3, λ; S3) exists if and onlyif λ ≡ O (mod 6 ) and λv(v - 1) ≡ O(mod 24); (ii) if G is one of the groups D4, Q4, and Z4 x Z2, a GBRD (v, 3,λ; G) exists if and only if λ ≡ O(mod 8) and λv(v - 1) ≡ O(mod 6); (iii) a GBRD (v, 3, λ; D6) exists if and onlyif λ ≡ O(mod 12). From these designs,families of regular group divisible designs are constructed.

DisciplinesPhysical Sciences and Mathematics

Publication DetailsPalmer, WD and Seberry, J, Bhaskar Rao designs over small groups, Ars Combinatoria, 26A, 1988, 125-148.

This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1033

Abstract:

BHASKAR RAO DESIGNS OVER SMALL GROUPS

William D. Palmer

Department of Pure Mathematics

University of Sydney

Australia

Jennifer Seberry

Department of Computer Science

Australian Defence Force Academy

Canberra

Australia.

We show that for each of the groups S3' D4 , °4 , Z4 x Z2

and 0 6 the necessary conditions are sufficient for

the existence of a generalized Bhaskar Rao design.

That is, we show that:

(i) a GBRp (v, " " 5, ) exists if and only

if , - O(Il)od 6 ) and ~v(v - 1) - O(mod 24)1

(ii) if G is one of the groups D" Q" and " x

a GBRD{v, , , " G) exists if and only if , '" O{mod 8 ) and AV(v - 1) - O(mod 6);

(iii) a GBRD (v, " " D6 ) exists if and only if , - O(mod 12 ).

From these designs,families of regular group divisible

designs are constructed.

ARS COMBINATORIA 26A(I988), pp. 125-148

'2'

1. Introduction

A desig~ is a pair (X,B) where X is a finite set of

eleme~ts and B is a collection of (not necessarily

distinct) subsets Bi (called blocks) of X.

A balanced incomplete block design, BIBD(v,b,r,k,X),

is an arrangement of v elements into b blocks such that:

(i) each element appears in exactly r blocks~

(ii) each block contains exactly k«v) elements~ and

(iii) each pair of distinct elements appear together

in exactly X blocks.

As r(k-l) = X{v-l) and vr = bk are well known

necessary conditions for the existence of a BIBD(v,b,r,

k,xl we denote this design by BIBD(v,k,X).

Let v and X be positive integers and K a set of

positive integers. An arrangement of the elements of a

set X into blocks is a pairwise balanced design,

PBD(v~K;X), if:

(il X contains exactly v elements;

(ii) if a block contains k elements then k belongs

to K;

(iii) each pair of distinct elements appear together

in exactly X blocks.

A pairwise balanced design PBD{v;{k};Xl, where

K = {k} consists of exactly one integer, is a BIBD(v,k,X).

It is well known that a PBD{v-I;{k,k-l};X) can be

obtained from the BIBD{v,b,r,k,X).

126

A generalized Bhaskar Rao design,- W is defined as

follows. Let W be a v x b matrix with entries from

Gu {o) where G = ( hI = e,h2, •• ~.hg} is a finite group

of order g. W is then expressed as a sum W = hlAI

+ ••• + hgAg' where AI ••••• Ag are v x b {O,l} - matrices

such that the Hadamard product A. * A. = 0 for any . + ~] -1 ~ r j. Denote by W the transpose of hI A, + ••• + -1

h A and let N = Al + ••• + A. In this paper we are g g . 9

concerned with the special case where W, denoted by

GBRD(v.b.r.k.A~G), satisfies

(i)

(ii)

'" reI + ~ (h, + ••• + g. (r-X) 1+ AJ.

It can be seen that the second condition requires

that N be the incidence matrix of a BIBD(v,b,r,k.A)

and thus we can use the shorter notation GBRD{v,k,A/G}

for a generalized Bhaskar Rao design. A GBRD(v,k,A;Z2)

will be sometimes referred to as a BRD{v.k.A).

A GBRD{v,k,A:G) with v '" b is a symmetric GBRD or

generalized weighing matrix. If W has no 0 entries the

GBRD is also known as a generalized Hadamard matrix.

A group divisible design. GOD, on v points is a

triple (X,S,A) where

(i) X is a set (of points), (ii) S is a class of non-empty subsets of x (called

groups) which partition X,

(iii) A is a class of subsets of X (called blocks),

each containing at least two points,

(iv) no block meets a group in more than one point,

(v) each pair {x,y) of points not contained in a

group is contained in precisely X blocks.

127

Bhaskar Rao designs with elements O,-~l have been

studied by a number of authors including Bhaskar Rao

(1966, 1970), Seberry (1982). 1984), Singh fl982),

Sinha (1978), Street (1981), Street and Rodger (1980)

and Vyas(1982). Bhaskar Rao (1966) used these designs

to construct partially balanced designs and this was

improved by Street and Rodger (1980) and Seberry (1984).

Another technique for studying partially balanced

designs has involved looking at generalized orthogonal

matrices which have elements from elementary abelian

groups and the element O. Matrices with group elements

as entries have been studied by Berman (1977, 1978),

Butson (1962, 1963), Delsarte and Goethals (1969),

Drake (1979), Rajkundlia (1978), Seberry (1979, 1980),

Shrikhande (1964) and Street (1979).

Generalized Hadamard matrices has been studied by

Street (1979), Seberry (1979, 1980), Dawson (1985), and

de Launey (1984, 1986).

Bhaskar Rao designs over elementary abelian groups

other than Z2 have been studied recently by Lam and

Seberry (1984) and Seberry (1985) .de Launey, Sarvate

and Seberry (1985) considered Bhaskar Rao designs over

Z4 which is an abelian (but not elementary) group.

Some Bhaskar Rao designs over the non-abelian groups

53 and 04 have been studied by Gibbons and Mathon

(1987).

In this paper we are concerned with generalized

Bhaskar Rao designs over the non-abelian groups

128

We use the following notation for initial blocks

of a GBRD. We

when the latin

say (au.bS

' ., •• Cyl is an initial block.

letters are developed mod n and the greek

subscripts are the elements of the group. which will

be placed in the incidence matrix in the position

indicated by the latin letter. That is in the

(i,b-l+ilth position we place p and so on.

We form the difference table of an initial block

(8 u .bS, •.•• cy ) by placing in the position headed by

" and by row y the element (x-y)o -1 where (x-y) is -1 n n

mod nand 6n is in the group.

A set of initial blocks will be said to form a

GBR difference set (if there is one initial block)

or GBR supplementary difference sets{if more than one)

if in the totality of elements

{X-ylon-l (mod n, G)

each non-zero element a g , a{mod nl, g E G, occurs A/IGI times.

Examples of the use of these GBR supplementary

difference sets are given in Seberry (1982).

2. Construction Theorems

The following theorem which was established in

Lam and Seberry -( 1984) and the associated corollaries

w1ll be extensively used in the remaining sections of

the paper.

Theorem 2.1

Suppose we ha',e a GBRD(v,Jc.)..;H) and a GBRD(k,j,v,;G).

Then there exists a GBRD(v,j.)..v,; H x GJ.

129

CoroLLary 2.2 (Lam and Seberry (1984))

Suppose we have Ii Br8n(.v,k,J...) and a GBRD(k,j,\J;G}.

Then there exists a GBRD(v,j,J...~iG}.

Corollary 2.3 {de Launey and Seberry (1982, 1984))

Suppose there exists Ii pairwise balanced design

PBD(v;X;J...} where X - {kl

, ... ,kb

} and Ii GBRD (ki,j,~;G)

for each k. £ X, then there exists a GBRD(v,j,J...~;G}. ,

3. Existence of GBRD in S3

The symmetric group, S3' can be generated by means

,- of the defining relations

(3.1 )

We firstly establish the necessary conditions for

the existence of GBRD over S3 with block size 3.

For a GBRD(v,3,J...:S3 } to exist

J...EO(mod6)

and there must exist a BIBD(v,3,J...).

(3.2)

We, therefore,

have the necessary conditions for the existence of a

GBRD{v,3,J...;S3)

v ) 3

J...(v-l} -' O{mod 2)

Av(v-l) _ O{mod 6)

(3.3)

(3.4)

( 3.5)

Gibbons and Mathon (1987) have proved the following

Theorem 3.1

Let W be Ii GBRD(v,k,A;G} and suppose that G contains

Ii normal subgroup T, ~ben there exists a GBRD(v,k, J...;H} ,

where H ~ GIT is the factor group of G with respect to

T.

130

It shoul~ be noted that this theorem is a genera­

lisation of Proposition 4.1 appearing in Lam and

5eberry (1984).

We now derive an extra necessary condition for

the existence of a GBRD(v,3'A; S3)'

Theorem 3.2

A necessary condition for the existence of a

GBRD(V,k,A;S3) is the existence of a BRD(v,k,A).

Proof: As Z2 is isomorphic to the factor group

S3/{l,p,p2} of {l,p,p2} in 53' we see that

a GBRD{v,k,A;S3) exists only if a GBRD

(V,k,A;Z2) - which is merely a BRD{v,k,A) -

exists.

Corollary 3.3

A OBRD(v,3,A; 83

) exists only when Av(v-l) = 0

(mod 24) (3.6)

Proof; By Theorem 1 of Seberry (1984) the BRD(v,3,A)

can only exist when Av(v-1) = O(mod 24).

Remark 3.4

As an immediate application we see that a GBRD

(3,3,6; S3) does not exist; and hence, for h,3, a GBRD(h,h,6,S3) does not exist because if it did

exist any three, distinct rows could be taken to be

a GBRD{3,3,6;S31.

131

Theorem 3.5

and

A GBRD(v,3,)";S3) exists if and only if

1 ~ O(mod 6), V) 3

lv(v-l) ~ O(mod 24), v) 3.

(3.7)

(3.8 )

Proof: It is clear that (3.7) and (3.8) are necessary

conditions for the existence of a GBRD(v.3.1;S3}'

To prove that (3.7) and (3.8) imply that a GBRD

(v,3,1:S3) exists, we put A ~ 6t and observe

that (3.7) and (3.8) are equivalent to one of

the following conditions

(a) t _ O(mod 2), V) 3

(b) t _ l(mod 2), v;;:: 0 or l(mod 4).

Case 1: t ~ l(mod 2). By applying Corollary 2.2 with

Hanani's theorem (Hall(1967), Lemma 15.5.1)

we only have to establish the existence of

GBRD(v,3,6;S3J for v f: K4 = {4.5,8,9.12} to

demonstrate the existence of all GBRD{u,3.6;S3}

for u ~ 0 or l(mod 4), U )-4. The GBRD

(v.3,6;S3) for v f: K4 are given in theA.ppendix.

Now by taking t copies (t odd) of the GBRD

(u,3.6;S3) whose existence has been established

we have shown the existence of GBRD(u.3,6t;S3)

for t ;: l{mod 2} and U ~ 0 or l(mod 4).

Case 2: t = 25 ~ O(mod}. By Corollary 2.2 and Hanani's

theorem {Hall(1967}, Lemma 15.4.2) we need to

establish the existence of GBRD(v,3,l2;S3J

for v f: {3,4,5,6,8,ll,14} to show that GBRD

(v,3,12)S3) exist for all v r 3. The designs

GBRD(v,3,12;S31 for v f: {3,6,14) are given in

the Appendix and the designs GBRD{v,3,12;S3)'

where v f: {4,S.8,11} exist,

132

v Construction

4 2 copies of the design with ; · 6

5 2 copies of the design with ; · 6

8 2 copies of the design with ; · 6

11 U,e a BrBD(H,S,ll with the GBRD

(S.3,6;S3 J

Thus we can see that a GBRD(v,3,12;S3) exists for

all v ) 3. Taking s copies of each of these designs

we obtain a GBRD(v.3.6(2s);S3J for all v ~ 3.

Thus, the theorem is established.

The quaternion group. Q4' consists of elements

l,-l,i,-i,j,-j,k,-k subject to the relations

i' j' . k' . -1,

ij • -ji k,

jk -kj i,

ki -ik . j. The dihedral group, D

4, can be generated by means

of the defining relations

r~ = a 2 = 1, ar = r 3 a

For the direct product, 22 x Z4' of grcups Z2 and

Z4 we represent the elements by 0,1,2,3,-0,-1,-2, and

-3.

We establish the necessary conditions for the

existence of a GBRD{v,3,A1G) where G is one of the

groups Q4,04' or Z2 x Z4'

133

For a GBRD(v,3,A;G) to exist

A"O{mod8) (4.1)

and there must exist a BIBD(v,3,A). We, therefore,

have the necessary conditions for the existence of a

GBRD(v,3,A;G)

v:7 3

A{v-l):o 0 (mod 2)

Av(v-l)" 0 (mod 6)

(4.2 )

(4.3)

(4.4 )

Theorem 3.1 reveals no further necessary conditions.

We consider A '" St, We see that:

(a) for t " 1,2,4 or 5{mod 6) that v _ 0 or

l(mod 3)

and (b) for t ~ O(mod 3) that v ) 3 are necessary

conditions for the existence of a GBRD

(v,3,St,G).

Theorem 4.1

The necessary conditions:

A _ 0 (mod 8)

Av(v-l) '" 0 (mod 6)

are sufficient for the existence of a GBRD(v,3,8t;G),

v ~ 3, t ~ 1,2,4,5 (mod 6).

Proof; GBRD(v,3,8;G) for v £ {3,4.6} are given in

the Appendix, Hence by Corollary 2.2 and Hanani's theorem (Ha11(1967), Lemma 15.4

GBRD{v,3,8;G) exist if v ~ 0 or l(mod 3).

134

Theorem 4.2

Tbe necossar9 conditions:

t ~ 0 (mod 3), v ~ 3

are sUfficient for tbe existence of a GBRD(v,3,8t;G).

Proof: By Hanani's theorem (Hal1(1967), Lemma 15.4.2)

and Corollary 2.2 we have to establish the

existence of GBRO{v,3,24;G) for v ~ {3,4,5,6,

8,11,14}, and these designs exist for the

following reasons:

v Construction

3 3 copies of the design with ; · 8 , 3 copies of the design with ; · 8

5 Use a B180(5,3,3) with the GBRD

(3,3,8;G)

6 3 copies of the design with ). · 8

7 Use a BIBO(7,3,3) with the GBRO

(3,3,.8;G)

8 Use a BI80(8,4,3) with the GBRD

(4,3,8;G)

11 Use a BI80(11,3,3) with the GBRO

(3,3,8;G)

14 Use the S81BO(15,7,3) to obtain a

PBO(14;{7,6};3)

Now the GBRO(v,3,8;G), v , {6,7} are

used.

Multiple ccpies of the GBRO{v,3,24;G) v } 3 give

the complete result.

Thus we have established the following theorem:

135

Theorem 4.3

The necessary conditions:

A _ 0 (mod 8)

Av(v-l} _ 0 (mod 6)

are sUfficient for the existence of a GBRD(v,3,A;G}.

5. Existence of GBRD in D6

The dihedral group, D6

, is given by the defining

relations

For a GBRD{v,3,X,D6 J to exist

A :: 0 (mod 12)

and there must exist a SISD(v,3,A')'

have the necessary conditions

v ? 3

X{v-l) =:0 (mod 2)

Av(v-l) :: 0 (mod 6)

(5.1 )

We therefore,

(5.2)

(5.3)

{5.4}

Theorem 3.1 reveals no further necessary condi-

tiona. It is clear that the conditions (5.3) and (5.4)

are redundant so that a GBRD(v,3,XfD61 exists only

if v> 3 and X :: O(mod 12).

LeDma. 5.1

A GBRD(S,3,12;D6 J exists.

Proof; The development of the initial blocks

(01,1 l ,3p ,4a ), (Ol,lp2a ,3 l ,4p2 ) (mod 5,63 )

is a GBRD(5,4,6;S3 1 • Applying Theorem 2.1.

with this design and a GBRD{4,3,2,Z2)(which

136

is known to exist) we can construct a GBRD

(S,3,12;S3 x Z2)' As 53 x Z2 is isomorphic

to D6 we have shown that a GBRD{5.3,12;D6 )

exists.

Theorem 5.2

The necessarg conditions:

X= a (iliad l2)

v ~ 3

are sufficient for the existence of a GBRD(v,3,X;D 6 }.

Proof: By Lemma 5.1 GBRD(5.3.12;D6 ) exists and the

designs GBRD{v,3.12;D6 }. v E {3,4,6,8,11,14}

are exhibited in the Appendix. Hence, by

Hanani's theorem (Hall{1967), Lemma 1S.4.2},

a GBRD{v,3,12;D6 ) exists for all v? 3; and

by taking multiples of these designs we can

construct all GBRD{V,3,X;D6 } for A =0 {mod 12},

V) 3.

6. Application

Proceeding as in Street and Rodger(1980), Lam and

Seberry (1984). Seberry (1982, 1984, 1985) and Gibbons

and Mathon (19B7), we now have the results;

Theorem 6.1

Whenever A = a(mod 6} and Av(v-l) = a (mod 24)

there exists a regular qro,Up divisible design ",jth

parameters

(6v, Xv(v-l), ~ 6,IR",v,n

'lbeonm 6.2

6).

Whenever A = o (IRod 8) and Av(v-1} = O(mod 6) there

exists a regular group divisib1e design with parameters 4Av(v-l} Mv-l) l _ 0 ' _ 1, _

(8v, 3 2' 3, "1 - • "'2 - 8' III - v. n '" 8)

137

Theorem 6.3

Whenever A = O(mod 12} tbere exists a regular group

divisible design with parameters

(12v, 2Av(v-I) ,

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Ber1iri-Heidelberg-New York, 238-245.

R. Vyas (1982), Some Bhaskar Rao Designs and applications

for k .. 3, A = 4, Universit!l of Indore J. Science_7, 16-25.

R.M. Wilson (1974), A few more squares. Inl Proc. Fifth

South Eastern Conf. on Cambinatorics, Graph Theory and

Computing, Congresses Numerantium XXI. Utilitas

Mathematica, Winnipeg, 675-660.

R.M. Wilson (1975), Construction and uses and pairwise

balanced designs, Combinatorics. Edited by M. Hall Jr.

and J.H. van Lint, (Mathematisch Centrum. Amsterdam).

19-42.

R.M. Wilson (I97S), An existence theory of pairwise

bala,nced designs III: proof of the existence conjectures,

J. Combinatorial Theory Series A, 18, 71-79.

142

Appendix

The designs used in the proofs of our theorems are

exhibited here. In many cases only the initial blocks are

shown:

1.

1

1

1

0

0

Tbe Group 53

GBRD (4.3.6;5 3 )

~ 1 1 1 1 1 1 1 1 0 0

~ P p2 0 pa p'" 0 0 0 1 1 2 0 0 0 "", p p 0 po a po

0 0 1 P rJ a p'" pa 1 p p2

It should be noted that this design was constructed

independently by Mathon and Gibbons (1987).

GBRD (5,3,6;5 3 )

1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

P p2 0 po p'" 0 0 0 0 0 0 0 0 1 1 1 1 1 2 0 0 0 0 0 0 P P 0 p. "", 1 1 0 0 0 po p2

0 0 0 1 P p2 0 0 0 p'" 1 p2 0 0 2 po 0 p po

0 1 p 0 0 "", po rJ 0 0 0 p2 0 p'" 0 2 1 0 p

GBRD {8,3.6;S31

(O>I,Opll,lp)' (<».,3pza.Opa)' (0)I,II,3 a J. (OI.lp' -3 a )

(Ol,la'~pz)' (OI,lpza,3pa)' (OI.I •• 3 1 ), (OI.lpa,3pza J

{mod 7, 53"

143

0

1

,Jo

1

0

GBRD (9.3,6;S~)

eeeee

eeeee o eeeee

eeeee

x. x. • x •

Y • Y' x. x.

Y • Y' • tx 0

t x tx t Y Y t yt t Y Y t Y t

wlEre D is a GBRO (4,3,6;S3)' and

e = (1,1,1)

Y'"' (l,p2,p),

so that

(l,pl,p)

(p,p2,11

x (l,p,p2) s '"' (a,p 2 a,pa)

t .. (p 2 a,a,pa)

xt- 1 '"' (p2 a,pa,a)

sy-l _(a,pa,p2a).

This design contains a subdesign on 4 points.

GBRO (12,3,6;8 3 )

{"'I ,Op,2al • ("'1 ,Op-l,5pal. ("'1,0 1.5p2al • (0 1,3 J. Sp).

(OI.11,5pa),(Ol.3p,8pl,(OI.41.5p2),(OI.4p.7a)'

(OI.2p2a.911,(OI,3a,4p21,(OI,lp,3p2al.(Ol,la.lOpal (mod II,S,)

GBRO (3,3,12,.8 3 )

U 1 1 1 1

P p' • p' p' p 1 P

1 1

p'. 1

rl •

1 1 1

P rl a

pa ta a

144

~] 1

p'

rJ>.

GBRD (6.3,12~S3)

(01.31 .2 a ). (01.lp.4pl. (01.41 .lpa1. (01 .2p .3p l. (0 I .lp'''_a.4pl

(0 I .2pa.3p2al. ("'1 .Op.3pa )' ("'1. 0 I' 1a ). (""I .Op2,4p2a )·

(.""I.Op2,la).(OOI'0pa,2p2al. (""1, 0 1 , 3p )' (mod 5, S,).

GBRD (14.3,12:S 3 )

(OI,ia ,(13-i)p 2a) i 3, ... ,12

{Opjp2.03-j)p21. j 1 •.•.• 6

(Ol'jpa,(13-j)pa l. j 1 ••.. ,6

(""1 .Op,lpa 1, (0)1 .Op2,2a1. (<»1 ,0 I .12 a ), (001 ,Op2a.11pl

(OOI,Op2a,11pa)'(0>1.01,9p2) (mod 13.S 3 1

2. The Group Z2 x Z~

We use 0.1.2,3.-0.-1,-2,-3 for the group elements of

Z2 x Z~ and * in the other positions in the design.

The following designs eXist;

GBRD (3,3,8;Z" x Z, )

[~ 0 0 0 0 0 0

-~1 1 2 3 -0 -1 -2

3 -3 -2 -1 -0 2

GBRD (4,3,8:Z" x Z~)

[~ 0 0 0 0 0 0 0 0 0 0 0 , , ,

~ ] 1 .,

-0 -2 ' -1 2 , 3 -3 , 0 3 -2

0 1 • -0 -2 • -1 2 • 3 -3 -2 0 3

• 0 -2 ' -0 2 ' -1 -3 • 3 3 -2 0

GBRD (6,3,8;Z" x Z~)

(mod 5, Z2 x Z~J

145

3. The Group D ..

GBRD (3,3,8; D, )

[ ~ 1 1 1 1 1 1

r~a 1 r r' r' a ra r'a

r' a r'a r' r'a r ra

GBRD (4.3.8~D .. )

[f 1 1 1 1 1 1 1 1 1 1 1 0 0 0

~l r' r3 3 r". 0 0 0 0 1 1 1 r a ra ra

r' a ,1!, 0 0 0 0 r3 r"a r ra r\. ?- • ;~ 0 0 0 ,1!, r' 1 r3 r 3 • r"a r3 a ra ra ra

GBRD (6,3,8, D.)

("'1'0 l,lr2). ("'I.Or.lal. CC>1.0r3,2ral. (""'1,Or2a,2r3a).

(0 1,2r3a,3r ). (0 1 ,2r2 • 3ral. (0 l,lr3 ,4r 3),

(Ol.lr2a,4 1 ) (mod 5,0 .. )

4. The Group Q ..

GBRD (3,3,8;Q .. )

[~ 1 1 1 1 1 1 -q i -k j -1 -j k

j k -1 i -i -j -k

GBRD (4,3,8;O.J

1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

1 j k -k -i -1 -j i 0 0 0 0 1 1 1 1

1 -i -j i 0 0 0 0 j -k -1 k j -k -1 -i

0 0 0 0 i -i -1 -k -j 1 j k -k -i k 1

146

GBRD (6,3,8;Q4)

(~1,li,4j)'{~l,l_k,4_i)'{~1,21,3k)'{~1,2_1,3_j)

(Ol·1_1,4j),{Ol,11,4_i),{Ol,21,3_kl,(Ol,2_i,3_1)

(mod 5,Q4).

5, The Group D6

GBRD (3,3,12;D6 )

(Ol,ll·2rl,{Ol·1a·2r2),{Ol,IrJ,2ra)·(Ol·lr2.2{3al

(mod 3,D6 )

GBRD (4.3,12;D6 )

(~1·11,2rl·(~1·lr2,2r~a)·{~1,lrJ·2rsal·{~l,lr~,2ral

(~l,lr2a·2rs)·{~l,lrJa,2a)

GBRO {6,3,12;D61

(~1·01,3r)'(~1,Or3,3ral·(=1,Or5,3r2a)

(~1,Oa·4r2}·(~1,Or~,4r3al'(~l,Or~a,4rsal

(01·21,3r21.(01,2r2a·3r3a),(Ol,2rJ·3al

(01.1rJ,4r21.(01,la,4r~a),(01.11,4r5al (mod 5.06)

(GBRD (8,3,12:D6 )

(=1'°1 ,5r l. (=1·0r2,3rlta I. (~l,Or3 .6r 5a I. ("'1 ,Or ~,5ra I,

(~1·0r2a·lr5J'("'l,Or3a·2a)

(01,31 ,4r21, (0 1 ,4r3 , 3r2 1, (0 1 ,5ra .2r l. (0 •• 6r5a.1r3a I

(0.,1I,6a)·(01.2i.5r~a),(01.3r·4r3al

(01,lr3.6rl.(Oi,2rJa,5r2a),(01,3ra,4r5al(mod 7,0&)

147

'-~'-'--'~6'

{=l,Or2a,lrs},{ml,Or\,5ra},(ml,Or3,7a}'(=1,Or3a ,6r sa }

(m l 'OI,8r ), (ml ,Or2,lOr Qa)

(Ol,ll,12r),(Ol,lr,12rsa),{Ol,lr2,12rJa),(Ol,lrJ,12r~a)

(Ol,21,llr2),{Ol,2r,llr~a),(Ol,2r3,11r3a),(Ol,2r2,11r2a)'

(Ol,31,lOrs),{Ol,3r2,lOr5a),(Ol,3r3,lOr3a),(Ol,3r~,10ral,

(Ol,41,9r3),(Ol,4r,9r3a),(Ol,4r2,9ra),(Ol,4rs,9r2a)'

(Ol,51,8rs),{Ol,5r2,8a)(Ol,5r2a,8r2)'

(Ol,6r3,7rl,(Ol,6r\a,7r2J,(Ol,61,7r2al (mod 13,D6 )

148

,

,