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## G = S32×C6order 216 = 23·33

### Direct product of C6, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C6
 Chief series C1 — C3 — C32 — C33 — S3×C32 — C3×S32 — S32×C6
 Lower central C32 — S32×C6
 Upper central C1 — C6

Generators and relations for S32×C6
G = < a,b,c,d,e | a6=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 492 in 162 conjugacy classes, 56 normal (12 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C22 [×7], S3 [×4], S3 [×6], C6, C6 [×2], C6 [×18], C23, C32, C32 [×2], C32 [×4], D6 [×2], D6 [×11], C2×C6 [×11], C3×S3 [×8], C3×S3 [×10], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×8], C22×S3 [×2], C22×C6, C33, S32 [×4], S3×C6 [×4], S3×C6 [×13], C2×C3⋊S3, C62 [×2], S3×C32 [×4], C3×C3⋊S3 [×2], C32×C6, C2×S32, S3×C2×C6 [×2], C3×S32 [×4], S3×C3×C6 [×2], C6×C3⋊S3, S32×C6
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×2], C6 [×7], C23, D6 [×6], C2×C6 [×7], C3×S3 [×2], C22×S3 [×2], C22×C6, S32, S3×C6 [×6], C2×S32, S3×C2×C6 [×2], C3×S32, S32×C6

Permutation representations of S32×C6
On 24 points - transitive group 24T547
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)

G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;

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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );

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)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)])

G:=TransitiveGroup(24,547);

S32×C6 is a maximal subgroup of   S32⋊Dic3  D64S32  D6⋊S32

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 6A 6B 6C ··· 6H 6I ··· 6P 6Q 6R 6S 6T ··· 6AE 6AF 6AG 6AH 6AI order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 6 6 6 ··· 6 6 ··· 6 6 6 6 6 ··· 6 6 6 6 6 size 1 1 3 3 3 3 9 9 1 1 2 ··· 2 4 4 4 1 1 2 ··· 2 3 ··· 3 4 4 4 6 ··· 6 9 9 9 9

54 irreducible representations

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

Matrix representation of S32×C6 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 6 6
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 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 1 0 0 0 0 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,1,6],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,1,0,0,0,0,1] >;

S32×C6 in GAP, Magma, Sage, TeX

S_3^2\times C_6
% in TeX

G:=Group("S3^2xC6");
// GroupNames label

G:=SmallGroup(216,170);
// by ID

G=gap.SmallGroup(216,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,730,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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