direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4×S32, C12⋊4D6, D6.8D6, Dic3⋊5D6, (S3×C12)⋊8C2, (C3×C12)⋊5C22, (S3×Dic3)⋊6C2, C6.D6⋊5C2, C6.7(C22×S3), (C3×C6).7C23, C32⋊1(C22×C4), (S3×C6).8C22, C3⋊Dic3⋊2C22, (C3×Dic3)⋊5C22, C3⋊1(S3×C2×C4), C2.1(C2×S32), (C4×C3⋊S3)⋊7C2, C3⋊S3⋊1(C2×C4), (C2×S32).2C2, (C3×S3)⋊1(C2×C4), (C2×C3⋊S3).14C22, SmallGroup(144,143)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4×S32 |
Generators and relations for C4×S32
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, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C2×C4, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C4×S32
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C2×S32, C4×S32
►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);
C4×S32 is a maximal subgroup of
S32⋊C8 C24⋊D6 S32⋊Q8 S32⋊D4 D12⋊23D6 Dic6⋊12D6 D12⋊15D6 Dic3⋊6S32
C4×S32 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 | C4×S3 | S32 | C2×S32 | C4×S32 |
| kernel | C4×S32 | S3×Dic3 | C6.D6 | S3×C12 | C4×C3⋊S3 | C2×S32 | S32 | C4×S3 | 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 C4×S32 ►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] >;
C4×S32 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