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University of WollongongResearch Online
Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences
1988
Bhaskar Rao designs over small groupsWilliam D. Palmer
Jennifer SeberryUniversity of Wollongong, [email protected]
Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:[email protected]
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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Lecture Notes in Mathematics, Vol 829, Springer-Verlag,
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
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and J.H. van Lint, (Mathematisch Centrum. Amsterdam).
19-42.
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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
,
,