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G = C6×S3≀C2order 432 = 24·33

Direct product of C6 and S3≀C2

direct product, non-abelian, soluble, monomial

Aliases: C6×S3≀C2, C32⋊(C6×D4), C332(C2×D4), (C32×C6)⋊1D4, S32⋊(C2×C6), C3⋊S3⋊(C3×D4), (C3×C6)⋊(C3×D4), (S32×C6)⋊9C2, (C2×S32)⋊5C6, C32⋊C4⋊(C2×C6), (C3×C3⋊S3)⋊1D4, (C3×S32)⋊2C22, (C6×C32⋊C4)⋊7C2, (C2×C32⋊C4)⋊3C6, C3⋊S3.1(C22×C6), (C3×C3⋊S3).3C23, (C3×C32⋊C4)⋊4C22, (C6×C3⋊S3).33C22, (C2×C3⋊S3).9(C2×C6), SmallGroup(432,754)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C6×S3≀C2
C1C32C3⋊S3C3×C3⋊S3C3×S32C3×S3≀C2 — C6×S3≀C2
C32C3⋊S3 — C6×S3≀C2
C1C6

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 2A2B2C2D2E2F2G3A3B3C···3H4A4B6A6B6C···6H6I···6P6Q6R6S6T6U···6AF12A12B12C12D
order12222222333···344666···66···666666···612121212
size11666699114···41818114···46···6999912···1218181818

54 irreducible representations

dim1111111122224444
type++++++++
imageC1C2C2C2C3C6C6C6D4D4C3×D4C3×D4S3≀C2C2×S3≀C2C3×S3≀C2C6×S3≀C2
kernelC6×S3≀C2C3×S3≀C2C6×C32⋊C4S32×C6C2×S3≀C2S3≀C2C2×C32⋊C4C2×S32C3×C3⋊S3C32×C6C3⋊S3C3×C6C6C3C2C1
# reps1412282411224488

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

3000
0300
0030
0003
,
3551
3115
6463
1665
,
4250
2352
2262
2432
,
2403
2514
1423
1635
,
3551
6213
2561
1665
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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