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## G = S32×C32order 324 = 22·34

### Direct product of C32, S3 and S3

Aliases: S32×C32, C3318D6, C341C22, C322C62, C336(C2×C6), C329(S3×C6), (S3×C33)⋊1C2, (S3×C32)⋊3C6, C31(S3×C3×C6), (C3×S3)⋊(C3×C6), (C3×C3⋊S3)⋊3C6, C3⋊S32(C3×C6), (C32×C3⋊S3)⋊1C2, SmallGroup(324,165)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C32
 Chief series C1 — C3 — C32 — C33 — C34 — S3×C33 — S32×C32
 Lower central C32 — S32×C32
 Upper central C1 — C32

Generators and relations for S32×C32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 664 in 224 conjugacy classes, 60 normal (8 characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3×C6, C33, C33, S32, S3×C6, C62, S3×C32, S3×C32, C3×C3⋊S3, C32×C6, C34, C3×S32, S3×C3×C6, S3×C33, C32×C3⋊S3, S32×C32
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, S32, S3×C6, C62, S3×C32, C3×S32, S3×C3×C6, S32×C32

Smallest permutation representation of S32×C32
On 36 points
Generators in S36
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 15 10)(2 13 11)(3 14 12)(4 7 36)(5 8 34)(6 9 35)(16 19 22)(17 20 23)(18 21 24)(25 28 31)(26 29 32)(27 30 33)
(1 10 15)(2 11 13)(3 12 14)(4 7 36)(5 8 34)(6 9 35)(16 19 22)(17 20 23)(18 21 24)(25 31 28)(26 32 29)(27 33 30)
(1 21)(2 19)(3 20)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 18)(11 16)(12 17)(13 22)(14 23)(15 24)(25 34)(26 35)(27 36)
(1 15 10)(2 13 11)(3 14 12)(4 36 7)(5 34 8)(6 35 9)(16 19 22)(17 20 23)(18 21 24)(25 31 28)(26 32 29)(27 33 30)
(1 30)(2 28)(3 29)(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(10 27)(11 25)(12 26)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

G:=sub<Sym(36)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,15,10)(2,13,11)(3,14,12)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33), (1,10,15)(2,11,13)(3,12,14)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,21)(2,19)(3,20)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,18)(11,16)(12,17)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (1,15,10)(2,13,11)(3,14,12)(4,36,7)(5,34,8)(6,35,9)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,30)(2,28)(3,29)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,27)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,15,10)(2,13,11)(3,14,12)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33), (1,10,15)(2,11,13)(3,12,14)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,21)(2,19)(3,20)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,18)(11,16)(12,17)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (1,15,10)(2,13,11)(3,14,12)(4,36,7)(5,34,8)(6,35,9)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,30)(2,28)(3,29)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,27)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36) );

G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,15,10),(2,13,11),(3,14,12),(4,7,36),(5,8,34),(6,9,35),(16,19,22),(17,20,23),(18,21,24),(25,28,31),(26,29,32),(27,30,33)], [(1,10,15),(2,11,13),(3,12,14),(4,7,36),(5,8,34),(6,9,35),(16,19,22),(17,20,23),(18,21,24),(25,31,28),(26,32,29),(27,33,30)], [(1,21),(2,19),(3,20),(4,30),(5,28),(6,29),(7,33),(8,31),(9,32),(10,18),(11,16),(12,17),(13,22),(14,23),(15,24),(25,34),(26,35),(27,36)], [(1,15,10),(2,13,11),(3,14,12),(4,36,7),(5,34,8),(6,35,9),(16,19,22),(17,20,23),(18,21,24),(25,31,28),(26,32,29),(27,33,30)], [(1,30),(2,28),(3,29),(4,21),(5,19),(6,20),(7,24),(8,22),(9,23),(10,27),(11,25),(12,26),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)]])

81 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Z 3AA ··· 3AI 6A ··· 6P 6Q ··· 6AH 6AI ··· 6AP order 1 2 2 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 3 3 9 1 ··· 1 2 ··· 2 4 ··· 4 3 ··· 3 6 ··· 6 9 ··· 9

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S32 C3×S32 kernel S32×C32 S3×C33 C32×C3⋊S3 C3×S32 S3×C32 C3×C3⋊S3 S3×C32 C33 C3×S3 C32 C32 C3 # reps 1 2 1 8 16 8 2 2 16 16 1 8

Matrix representation of S32×C32 in GL4(𝔽7) generated by

 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 1 0 0 6 0
,
 1 0 0 0 0 1 0 0 0 0 0 6 0 0 6 0
,
 0 1 0 0 6 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 6 6 0 0 0 0 6 0 0 0 0 6
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,6,0],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,6,0,0,0,0,6] >;

S32×C32 in GAP, Magma, Sage, TeX

S_3^2\times C_3^2
% in TeX

G:=Group("S3^2xC3^2");
// GroupNames label

G:=SmallGroup(324,165);
// by ID

G=gap.SmallGroup(324,165);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1090,7781]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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