direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4xS32, C12:4D6, D6.8D6, Dic3:5D6, (S3xC12):8C2, (C3xC12):5C22, (S3xDic3):6C2, C6.D6:5C2, C6.7(C22xS3), (C3xC6).7C23, C32:1(C22xC4), (S3xC6).8C22, C3:Dic3:2C22, (C3xDic3):5C22, C3:1(S3xC2xC4), C2.1(C2xS32), (C4xC3:S3):7C2, C3:S3:1(C2xC4), (C2xS32).2C2, (C3xS3):1(C2xC4), (C2xC3:S3).14C22, SmallGroup(144,143)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4xS32 |
Generators and relations for C4xS32
G = < a,b,c,d,e | a4=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 352 in 116 conjugacy classes, 46 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2xC4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C22xC4, C3xS3, C3:S3, C3xC6, C4xS3, C4xS3, C2xDic3, C2xC12, C22xS3, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, S3xC2xC4, S3xDic3, C6.D6, S3xC12, C4xC3:S3, C2xS32, C4xS32
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S32, S3xC2xC4, C2xS32, C4xS32
►On 24 points - transitive group 24T224
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 24)(2 21)(3 22)(4 23)(5 20)(6 17)(7 18)(8 19)(9 14)(10 15)(11 16)(12 13)
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), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,24)(2,21)(3,22)(4,23)(5,20)(6,17)(7,18)(8,19)(9,14)(10,15)(11,16)(12,13)>;
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), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,24)(2,21)(3,22)(4,23)(5,20)(6,17)(7,18)(8,19)(9,14)(10,15)(11,16)(12,13) );
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)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,24),(2,21),(3,22),(4,23),(5,20),(6,17),(7,18),(8,19),(9,14),(10,15),(11,16),(12,13)]])
G:=TransitiveGroup(24,224);
C4xS32 is a maximal subgroup of
S32:C8 C24:D6 S32:Q8 S32:D4 D12:23D6 Dic6:12D6 D12:15D6 Dic3:6S32
C4xS32 is a maximal quotient of
C24:D6 C24.63D6 C24.64D6 C24.D6 C62.6C23 Dic3:5Dic6 C62.8C23 C62.47C23 C62.48C23 C62.49C23 Dic3:4D12 C62.51C23 C62.53C23 C62.72C23 C62.74C23 C62.91C23 Dic3:6S32
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | C4xS3 | S32 | C2xS32 | C4xS32 |
| kernel | C4xS32 | S3xDic3 | C6.D6 | S3xC12 | C4xC3:S3 | C2xS32 | S32 | C4xS3 | Dic3 | C12 | D6 | S3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C4xS32 ►in GL4(F5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 3 | 0 | 0 |
| 4 | 3 | 0 | 0 |
| 3 | 3 | 4 | 3 |
| 0 | 4 | 3 | 0 |
| 1 | 2 | 2 | 3 |
| 1 | 3 | 3 | 1 |
| 0 | 4 | 0 | 1 |
| 1 | 3 | 4 | 1 |
| 3 | 0 | 0 | 2 |
| 0 | 4 | 2 | 1 |
| 2 | 2 | 0 | 2 |
| 1 | 0 | 0 | 1 |
| 4 | 2 | 3 | 3 |
| 4 | 4 | 1 | 2 |
| 0 | 4 | 0 | 1 |
| 4 | 4 | 2 | 2 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,4,3,0,3,3,3,4,0,0,4,3,0,0,3,0],[1,1,0,1,2,3,4,3,2,3,0,4,3,1,1,1],[3,0,2,1,0,4,2,0,0,2,0,0,2,1,2,1],[4,4,0,4,2,4,4,4,3,1,0,2,3,2,1,2] >;
C4xS32 in GAP, Magma, Sage, TeX
C_4\times S_3^2
% in TeX
G:=Group("C4xS3^2"); // GroupNames label
G:=SmallGroup(144,143);
// by ID
G=gap.SmallGroup(144,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations