metabelian, supersoluble, monomial
Aliases: C24⋊6D6, D12⋊2D6, D24⋊6S3, Dic6⋊2D6, C8⋊3S32, C24⋊C2⋊3S3, (C3×D24)⋊1C2, C6.30(S3×D4), C24⋊S3⋊1C2, D6⋊D6⋊2C2, C3⋊2(Q8⋊3D6), C3⋊2(D8⋊S3), (C3×C24)⋊1C22, C32⋊2D8⋊3C2, D12⋊S3⋊1C2, (C3×D12)⋊4C22, C3⋊Dic3.13D4, C32⋊5(C8⋊C22), Dic6⋊S3⋊2C2, C2.7(D6⋊D6), C12.69(C22×S3), (C3×C12).46C23, C32⋊4C8⋊1C22, (C3×Dic6)⋊4C22, C4.67(C2×S32), (C3×C24⋊C2)⋊1C2, (C2×C3⋊S3).17D4, (C3×C6).30(C2×D4), (C4×C3⋊S3).9C22, SmallGroup(288,446)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊6D6
G = < a,b,c | a24=b6=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >
Subgroups: 722 in 146 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, D24, D4⋊S3, D4.S3, Q8⋊2S3, C3×D8, C3×SD16, S3×D4, D4⋊2S3, Q8⋊3S3, C32⋊4C8, C3×C24, S3×Dic3, D6⋊S3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, C2×S32, D8⋊S3, Q8⋊3D6, C32⋊2D8, Dic6⋊S3, C3×C24⋊C2, C3×D24, C24⋊S3, D12⋊S3, D6⋊D6, C24⋊6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S32, S3×D4, C2×S32, D8⋊S3, Q8⋊3D6, D6⋊D6, C24⋊6D6
►On 48 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 26 17 34 9 42)(2 25 18 33 10 41)(3 48 19 32 11 40)(4 47 20 31 12 39)(5 46 21 30 13 38)(6 45 22 29 14 37)(7 44 23 28 15 36)(8 43 24 27 16 35)
(1 17)(2 22)(4 8)(5 13)(6 18)(7 23)(10 14)(11 19)(12 24)(16 20)(25 45)(27 31)(28 36)(29 41)(30 46)(33 37)(34 42)(35 47)(39 43)(40 48)
G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,26,17,34,9,42)(2,25,18,33,10,41)(3,48,19,32,11,40)(4,47,20,31,12,39)(5,46,21,30,13,38)(6,45,22,29,14,37)(7,44,23,28,15,36)(8,43,24,27,16,35), (1,17)(2,22)(4,8)(5,13)(6,18)(7,23)(10,14)(11,19)(12,24)(16,20)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,26,17,34,9,42)(2,25,18,33,10,41)(3,48,19,32,11,40)(4,47,20,31,12,39)(5,46,21,30,13,38)(6,45,22,29,14,37)(7,44,23,28,15,36)(8,43,24,27,16,35), (1,17)(2,22)(4,8)(5,13)(6,18)(7,23)(10,14)(11,19)(12,24)(16,20)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48) );
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),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,26,17,34,9,42),(2,25,18,33,10,41),(3,48,19,32,11,40),(4,47,20,31,12,39),(5,46,21,30,13,38),(6,45,22,29,14,37),(7,44,23,28,15,36),(8,43,24,27,16,35)], [(1,17),(2,22),(4,8),(5,13),(6,18),(7,23),(10,14),(11,19),(12,24),(16,20),(25,45),(27,31),(28,36),(29,41),(30,46),(33,37),(34,42),(35,47),(39,43),(40,48)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 12 | 12 | 18 | 2 | 2 | 4 | 2 | 12 | 18 | 2 | 2 | 4 | 24 | 24 | 24 | 4 | 36 | 4 | 4 | 4 | 4 | 24 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C8⋊C22 | S32 | S3×D4 | C2×S32 | D8⋊S3 | Q8⋊3D6 | D6⋊D6 | C24⋊6D6 |
| kernel | C24⋊6D6 | C32⋊2D8 | Dic6⋊S3 | C3×C24⋊C2 | C3×D24 | C24⋊S3 | D12⋊S3 | D6⋊D6 | C24⋊C2 | D24 | C3⋊Dic3 | C2×C3⋊S3 | C24 | Dic6 | D12 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C24⋊6D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 2 |
| 1 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [0,0,1,0,0,4,0,3,3,0,1,0,0,1,0,4],[0,0,0,1,1,0,4,0,0,3,0,3,1,0,2,0],[1,0,2,0,0,4,0,0,0,0,4,0,0,0,0,1] >;
C24⋊6D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes_6D_6 % in TeX
G:=Group("C24:6D6"); // GroupNames label
G:=SmallGroup(288,446);
// by ID
G=gap.SmallGroup(288,446);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,58,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations