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## G = C32×C3⋊S3order 162 = 2·34

### Direct product of C32 and C3⋊S3

Aliases: C32×C3⋊S3, C32C2, C342C2, C337S3, C338C6, C3⋊(S3×C32), C324(C3×C6), C325(C3×S3), SmallGroup(162,52)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C3⋊S3
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 344 in 160 conjugacy classes, 42 normal (6 characteristic)
C1, C2, C3 [×8], C3 [×16], S3 [×4], C6 [×4], C32 [×2], C32 [×16], C32 [×56], C3×S3 [×16], C3⋊S3, C3×C6, C33 [×8], C33 [×16], S3×C32 [×4], C3×C3⋊S3 [×4], C34, C32×C3⋊S3
Quotients: C1, C2, C3 [×4], S3 [×4], C6 [×4], C32, C3×S3 [×16], C3⋊S3, C3×C6, S3×C32 [×4], C3×C3⋊S3 [×4], C32×C3⋊S3

Permutation representations of C32×C3⋊S3
On 18 points - transitive group 18T79
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 14 11)(2 15 12)(3 13 10)(4 9 16)(5 7 17)(6 8 18)
(1 10 15)(2 11 13)(3 12 14)(4 7 18)(5 8 16)(6 9 17)
(1 14 11)(2 15 12)(3 13 10)(4 16 9)(5 17 7)(6 18 8)
(1 4)(2 5)(3 6)(7 15)(8 13)(9 14)(10 18)(11 16)(12 17)

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

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,11)(2,15,12)(3,13,10)(4,9,16)(5,7,17)(6,8,18), (1,10,15)(2,11,13)(3,12,14)(4,7,18)(5,8,16)(6,9,17), (1,14,11)(2,15,12)(3,13,10)(4,16,9)(5,17,7)(6,18,8), (1,4)(2,5)(3,6)(7,15)(8,13)(9,14)(10,18)(11,16)(12,17) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,14,11),(2,15,12),(3,13,10),(4,9,16),(5,7,17),(6,8,18)], [(1,10,15),(2,11,13),(3,12,14),(4,7,18),(5,8,16),(6,9,17)], [(1,14,11),(2,15,12),(3,13,10),(4,16,9),(5,17,7),(6,18,8)], [(1,4),(2,5),(3,6),(7,15),(8,13),(9,14),(10,18),(11,16),(12,17)])

G:=TransitiveGroup(18,79);

C32×C3⋊S3 is a maximal subgroup of
C34⋊C4  S32×C32  C3317D6  C331D9  C34⋊C6  C34⋊S3  C34.C6  C34.S3  C343S3  C34.7S3  C345S3  C346S3
C32×C3⋊S3 is a maximal quotient of
3+ 1+4⋊C2  3+ 1+42C2  3- 1+4⋊C2  3- 1+42C2

54 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AR 6A ··· 6H order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9

54 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 S3 C3×S3 kernel C32×C3⋊S3 C34 C3×C3⋊S3 C33 C33 C32 # reps 1 1 8 8 4 32

Matrix representation of C32×C3⋊S3 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 2 0 0 0 0 2
,
 2 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 2 0 0 0 0 4 0 0 0 0 2 0 0 0 0 4
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[2,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,4,0,0,0,0,2,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C32×C3⋊S3 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes S_3
% in TeX

G:=Group("C3^2xC3:S3");
// GroupNames label

G:=SmallGroup(162,52);
// by ID

G=gap.SmallGroup(162,52);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,723,2704]);
// Polycyclic

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

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