metabelian, supersoluble, monomial
Aliases: C62⋊8D4, C62.123C23, (C2×C6)⋊5D12, C23.35S32, C6.74(S3×D4), (C2×Dic3)⋊5D6, C6.88(C2×D12), C32⋊6C22≀C2, (C22×S3)⋊4D6, C3⋊2(C23⋊2D6), C3⋊4(D6⋊D4), (C22×C6).81D6, C6.D4⋊14S3, (C6×Dic3)⋊4C22, C6.D12⋊11C2, C2.33(Dic3⋊D6), C22⋊5(C3⋊D12), (C2×C62).42C22, (C2×C3⋊S3)⋊15D4, (C2×C3⋊D4)⋊8S3, (S3×C2×C6)⋊3C22, (C6×C3⋊D4)⋊13C2, (C2×C6)⋊7(C3⋊D4), (C23×C3⋊S3)⋊1C2, C6.25(C2×C3⋊D4), C22.146(C2×S32), (C3×C6).169(C2×D4), (C2×C3⋊D12)⋊13C2, C2.26(C2×C3⋊D12), (C3×C6.D4)⋊10C2, (C2×C6).142(C22×S3), (C22×C3⋊S3).78C22, SmallGroup(288,629)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊8D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1634 in 327 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22≀C2, C3×Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C6.D4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C3⋊D12, C6×Dic3, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C22×C3⋊S3, C22×C3⋊S3, C2×C62, D6⋊D4, C23⋊2D6, C6.D12, C3×C6.D4, C2×C3⋊D12, C6×C3⋊D4, C23×C3⋊S3, C62⋊8D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C22≀C2, S32, C2×D12, S3×D4, C2×C3⋊D4, C3⋊D12, C2×S32, D6⋊D4, C23⋊2D6, C2×C3⋊D12, Dic3⋊D6, C62⋊8D4
►On 24 points - transitive group 24T672
(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 11 3 10 2 12)(4 8 6 7 5 9)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 15 6 24)(2 17 4 20)(3 13 5 22)(7 23 11 14)(8 19 12 16)(9 21 10 18)
(1 2)(4 6)(7 9)(10 11)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)
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), (1,11,3,10,2,12)(4,8,6,7,5,9)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,15,6,24)(2,17,4,20)(3,13,5,22)(7,23,11,14)(8,19,12,16)(9,21,10,18), (1,2)(4,6)(7,9)(10,11)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23)>;
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), (1,11,3,10,2,12)(4,8,6,7,5,9)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,15,6,24)(2,17,4,20)(3,13,5,22)(7,23,11,14)(8,19,12,16)(9,21,10,18), (1,2)(4,6)(7,9)(10,11)(13,22)(14,21)(15,20)(16,19)(17,24)(18,23) );
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)], [(1,11,3,10,2,12),(4,8,6,7,5,9),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,15,6,24),(2,17,4,20),(3,13,5,22),(7,23,11,14),(8,19,12,16),(9,21,10,18)], [(1,2),(4,6),(7,9),(10,11),(13,22),(14,21),(15,20),(16,19),(17,24),(18,23)]])
G:=TransitiveGroup(24,672);
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 6A | ··· | 6F | 6G | ··· | 6Q | 6R | 6S | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | C3⋊D4 | S32 | S3×D4 | C3⋊D12 | C2×S32 | Dic3⋊D6 |
| kernel | C62⋊8D4 | C6.D12 | C3×C6.D4 | C2×C3⋊D12 | C6×C3⋊D4 | C23×C3⋊S3 | C6.D4 | C2×C3⋊D4 | C2×C3⋊S3 | C62 | C2×Dic3 | C22×S3 | C22×C6 | C2×C6 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 2 | 4 | 4 | 1 | 4 | 2 | 1 | 4 |
Matrix representation of C62⋊8D4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
C62⋊8D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_8D_4
% in TeX
G:=Group("C6^2:8D4"); // GroupNames label
G:=SmallGroup(288,629);
// by ID
G=gap.SmallGroup(288,629);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations