direct product, non-abelian, soluble, monomial
Aliases: C3xS32:C4, S32:C12, C6.19S3wrC2, C6.D6:4C6, (C32xC6).1D4, C33:2(C22:C4), (C2xS32).C6, (C3xS32):1C4, (S32xC6).1C2, C2.1(C3xS3wrC2), (C2xC32:C4):1C6, (C6xC32:C4):2C2, C3:S3.2(C3xD4), (C3xC3:S3).5D4, (C3xC6).1(C3xD4), C32:(C3xC22:C4), C3:S3.2(C2xC12), (C3xC6.D6):7C2, (C6xC3:S3).4C22, (C3xC3:S3).4(C2xC4), (C2xC3:S3).4(C2xC6), SmallGroup(432,574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xS32:C4
G = < a,b,c,d,e,f | a3=b3=c2=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=fdf-1=b-1, bd=db, be=eb, fbf-1=ede=d-1, cd=dc, ce=ec, fcf-1=e, fef-1=c >
Subgroups: 652 in 132 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, D6, C2xC6, C22:C4, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, C2xC12, C22xS3, C22xC6, C33, C3xDic3, C3xC12, C32:C4, S32, S32, S3xC6, C2xC3:S3, C62, C3xC22:C4, S3xC32, C3xC3:S3, C32xC6, C6.D6, S3xC12, C2xC32:C4, C2xS32, S3xC2xC6, C32xDic3, C3xC32:C4, C3xS32, C3xS32, S3xC3xC6, C6xC3:S3, S32:C4, C3xC6.D6, C6xC32:C4, S32xC6, C3xS32:C4
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C12, C2xC6, C22:C4, C2xC12, C3xD4, C3xC22:C4, S3wrC2, S32:C4, C3xS3wrC2, C3xS32:C4
►On 24 points - transitive group 24T1317
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 21 15)(12 22 16)
(2 20 7)(4 18 5)(10 24 14)(12 22 16)
(2 12)(4 10)(5 24)(7 22)(14 18)(16 20)
(1 6 19)(3 8 17)(9 13 23)(11 15 21)
(1 11)(3 9)(6 21)(8 23)(13 17)(15 19)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
G:=sub<Sym(24)| (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,21,15)(12,22,16), (2,20,7)(4,18,5)(10,24,14)(12,22,16), (2,12)(4,10)(5,24)(7,22)(14,18)(16,20), (1,6,19)(3,8,17)(9,13,23)(11,15,21), (1,11)(3,9)(6,21)(8,23)(13,17)(15,19), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
G:=Group( (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,21,15)(12,22,16), (2,20,7)(4,18,5)(10,24,14)(12,22,16), (2,12)(4,10)(5,24)(7,22)(14,18)(16,20), (1,6,19)(3,8,17)(9,13,23)(11,15,21), (1,11)(3,9)(6,21)(8,23)(13,17)(15,19), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
G=PermutationGroup([[(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,13),(10,24,14),(11,21,15),(12,22,16)], [(2,20,7),(4,18,5),(10,24,14),(12,22,16)], [(2,12),(4,10),(5,24),(7,22),(14,18),(16,20)], [(1,6,19),(3,8,17),(9,13,23),(11,15,21)], [(1,11),(3,9),(6,21),(8,23),(13,17),(15,19)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
G:=TransitiveGroup(24,1317);
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 6Q | ··· | 6V | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 18 | 18 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 12 | ··· | 12 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 18 | 18 | 18 | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | C3xD4 | C3xD4 | S3wrC2 | S32:C4 | S32:C4 | C3xS3wrC2 | C3xS32:C4 |
| kernel | C3xS32:C4 | C3xC6.D6 | C6xC32:C4 | S32xC6 | S32:C4 | C3xS32 | C6.D6 | C2xC32:C4 | C2xS32 | S32 | C3xC3:S3 | C32xC6 | C3:S3 | C3xC6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of C3xS32:C4 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 5 | 5 | 1 |
| 3 | 1 | 1 | 5 |
| 6 | 4 | 6 | 3 |
| 1 | 6 | 6 | 5 |
| 6 | 4 | 0 | 6 |
| 5 | 5 | 0 | 6 |
| 2 | 3 | 1 | 1 |
| 6 | 2 | 0 | 4 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 5 | 1 | 6 | 2 |
| 6 | 6 | 2 | 3 |
| 6 | 5 | 3 | 3 |
| 2 | 4 | 3 | 2 |
| 3 | 4 | 4 | 3 |
| 4 | 1 | 3 | 6 |
| 4 | 0 | 4 | 3 |
| 0 | 4 | 3 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[6,5,2,6,4,5,3,2,0,0,1,0,6,6,1,4],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[5,6,6,2,1,6,5,4,6,2,3,3,2,3,3,2],[3,4,4,0,4,1,0,4,4,3,4,3,3,6,3,6] >;
C3xS32:C4 in GAP, Magma, Sage, TeX
C_3\times S_3^2\rtimes C_4
% in TeX
G:=Group("C3xS3^2:C4"); // GroupNames label
G:=SmallGroup(432,574);
// by ID
G=gap.SmallGroup(432,574);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,176,4037,3036,362,1189,1203]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^3=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=f*d*f^-1=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=e*d*e=d^-1,c*d=d*c,c*e=e*c,f*c*f^-1=e,f*e*f^-1=c>;
// generators/relations