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## G = C6×S3≀C2order 432 = 24·33

### Direct product of C6 and S3≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C6×S3≀C2
 Chief series C1 — C32 — C3⋊S3 — C3×C3⋊S3 — C3×S32 — C3×S3≀C2 — C6×S3≀C2
 Lower central C32 — C3⋊S3 — C6×S3≀C2
 Upper central C1 — C6

Generators and relations for C6×S3≀C2
G = < a,b,c,d,e | a6=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 956 in 192 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C4 [×2], C22 [×9], S3 [×8], C6, C6 [×18], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C12 [×2], D6 [×12], C2×C6 [×13], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3×C6, C3×C6 [×8], C2×C12, C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C33, C32⋊C4 [×2], S32 [×4], S32 [×2], S3×C6 [×16], C2×C3⋊S3, C62 [×2], C6×D4, S3×C32 [×4], C3×C3⋊S3 [×2], C32×C6, S3≀C2 [×4], C2×C32⋊C4, C2×S32 [×2], S3×C2×C6 [×2], C3×C32⋊C4 [×2], C3×S32 [×4], C3×S32 [×2], S3×C3×C6 [×2], C6×C3⋊S3, C2×S3≀C2, C3×S3≀C2 [×4], C6×C32⋊C4, S32×C6 [×2], C6×S3≀C2
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], C2×D4, C3×D4 [×2], C22×C6, C6×D4, S3≀C2, C2×S3≀C2, C3×S3≀C2, C6×S3≀C2

Permutation representations of C6×S3≀C2
On 24 points - transitive group 24T1316
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)
(7 9 11)(8 10 12)(19 23 21)(20 24 22)
(1 5 3)(2 6 4)(13 15 17)(14 16 18)
(1 21 16 11)(2 22 17 12)(3 23 18 7)(4 24 13 8)(5 19 14 9)(6 20 15 10)
(1 4)(2 5)(3 6)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(24,1316);

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 6I ··· 6P 6Q 6R 6S 6T 6U ··· 6AF 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 ··· 6 12 12 12 12 size 1 1 6 6 6 6 9 9 1 1 4 ··· 4 18 18 1 1 4 ··· 4 6 ··· 6 9 9 9 9 12 ··· 12 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 C3×D4 C3×D4 S3≀C2 C2×S3≀C2 C3×S3≀C2 C6×S3≀C2 kernel C6×S3≀C2 C3×S3≀C2 C6×C32⋊C4 S32×C6 C2×S3≀C2 S3≀C2 C2×C32⋊C4 C2×S32 C3×C3⋊S3 C32×C6 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 4 1 2 2 8 2 4 1 1 2 2 4 4 8 8

Matrix representation of C6×S3≀C2 in GL4(𝔽7) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 3 5 5 1 3 1 1 5 6 4 6 3 1 6 6 5
,
 4 2 5 0 2 3 5 2 2 2 6 2 2 4 3 2
,
 2 4 0 3 2 5 1 4 1 4 2 3 1 6 3 5
,
 3 5 5 1 6 2 1 3 2 5 6 1 1 6 6 5
G:=sub<GL(4,GF(7))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[2,2,1,1,4,5,4,6,0,1,2,3,3,4,3,5],[3,6,2,1,5,2,5,6,5,1,6,6,1,3,1,5] >;

C6×S3≀C2 in GAP, Magma, Sage, TeX

C_6\times S_3\wr C_2
% in TeX

G:=Group("C6xS3wrC2");
// GroupNames label

G:=SmallGroup(432,754);
// by ID

G=gap.SmallGroup(432,754);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,3,365,4037,3036,201,1189,622]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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