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G = C6xS3wrC2order 432 = 24·33

Direct product of C6 and S3wrC2

direct product, non-abelian, soluble, monomial

Aliases: C6xS3wrC2, C32:(C6xD4), C33:2(C2xD4), (C32xC6):1D4, S32:(C2xC6), C3:S3:(C3xD4), (C3xC6):(C3xD4), (S32xC6):9C2, (C2xS32):5C6, C32:C4:(C2xC6), (C3xC3:S3):1D4, (C3xS32):2C22, (C6xC32:C4):7C2, (C2xC32:C4):3C6, C3:S3.1(C22xC6), (C3xC3:S3).3C23, (C3xC32:C4):4C22, (C6xC3:S3).33C22, (C2xC3:S3).9(C2xC6), SmallGroup(432,754)

Series: Derived Chief Lower central Upper central

C1C32C3:S3 — C6xS3wrC2
C1C32C3:S3C3xC3:S3C3xS32C3xS3wrC2 — C6xS3wrC2
C32C3:S3 — C6xS3wrC2
C1C6

Generators and relations for C6xS3wrC2
 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, C3, C3, C4, C22, S3, C6, C6, C2xC4, D4, C23, C32, C32, C12, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C2xC12, C3xD4, C22xS3, C22xC6, C33, C32:C4, S32, S32, S3xC6, C2xC3:S3, C62, C6xD4, S3xC32, C3xC3:S3, C32xC6, S3wrC2, C2xC32:C4, C2xS32, S3xC2xC6, C3xC32:C4, C3xS32, C3xS32, S3xC3xC6, C6xC3:S3, C2xS3wrC2, C3xS3wrC2, C6xC32:C4, S32xC6, C6xS3wrC2
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C6xD4, S3wrC2, C2xS3wrC2, C3xS3wrC2, C6xS3wrC2

Permutation representations of C6xS3wrC2
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++++++++
imageC1C2C2C2C3C6C6C6D4D4C3xD4C3xD4S3wrC2C2xS3wrC2C3xS3wrC2C6xS3wrC2
kernelC6xS3wrC2C3xS3wrC2C6xC32:C4S32xC6C2xS3wrC2S3wrC2C2xC32:C4C2xS32C3xC3:S3C32xC6C3:S3C3xC6C6C3C2C1
# reps1412282411224488

Matrix representation of C6xS3wrC2 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] >;

C6xS3wrC2 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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