metabelian, supersoluble, monomial
Aliases: C24.64D6, C8.12S32, (S3×C8)⋊8S3, C3⋊C8.30D6, C8⋊S3⋊7S3, D6.3(C4×S3), (S3×C24)⋊14C2, C24⋊S3⋊8C2, C3⋊2(C8○D12), (C4×S3).29D6, C32⋊2(C8○D4), C3⋊1(D12.C4), C3⋊D12.1C4, D6⋊S3.1C4, Dic3.1(C4×S3), C32⋊2Q8.1C4, C12.29D6⋊9C2, D6.Dic3⋊12C2, (C3×C24).46C22, D6.D6.2C2, (S3×C12).39C22, (C3×C12).139C23, C12.138(C22×S3), C32⋊4C8.19C22, (S3×C3⋊C8)⋊8C2, C2.7(C4×S32), C6.5(S3×C2×C4), C4.85(C2×S32), (S3×C6).2(C2×C4), (C3×C8⋊S3)⋊11C2, (C3×C3⋊C8).24C22, (C3×C6).5(C22×C4), (C4×C3⋊S3).58C22, C3⋊Dic3.18(C2×C4), (C3×Dic3).12(C2×C4), (C2×C3⋊S3).14(C2×C4), SmallGroup(288,452)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.64D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=cac-1=a5, cbc-1=a12b-1 >
Subgroups: 434 in 134 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, S3×C8, C8⋊S3, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C4○D12, C3×C3⋊C8, C32⋊4C8, C3×C24, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, C8○D12, D12.C4, S3×C3⋊C8, C12.29D6, D6.Dic3, S3×C24, C3×C8⋊S3, C24⋊S3, D6.D6, C24.64D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S32, S3×C2×C4, C2×S32, C8○D12, D12.C4, C4×S32, C24.64D6
►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 35 17 43 9 27)(2 40 18 48 10 32)(3 45 19 29 11 37)(4 26 20 34 12 42)(5 31 21 39 13 47)(6 36 22 44 14 28)(7 41 23 25 15 33)(8 46 24 30 16 38)
(1 23 13 11)(2 4 14 16)(3 9 15 21)(5 19 17 7)(6 24 18 12)(8 10 20 22)(25 27 37 39)(26 32 38 44)(28 42 40 30)(29 47 41 35)(31 33 43 45)(34 48 46 36)
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), (1,35,17,43,9,27)(2,40,18,48,10,32)(3,45,19,29,11,37)(4,26,20,34,12,42)(5,31,21,39,13,47)(6,36,22,44,14,28)(7,41,23,25,15,33)(8,46,24,30,16,38), (1,23,13,11)(2,4,14,16)(3,9,15,21)(5,19,17,7)(6,24,18,12)(8,10,20,22)(25,27,37,39)(26,32,38,44)(28,42,40,30)(29,47,41,35)(31,33,43,45)(34,48,46,36)>;
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), (1,35,17,43,9,27)(2,40,18,48,10,32)(3,45,19,29,11,37)(4,26,20,34,12,42)(5,31,21,39,13,47)(6,36,22,44,14,28)(7,41,23,25,15,33)(8,46,24,30,16,38), (1,23,13,11)(2,4,14,16)(3,9,15,21)(5,19,17,7)(6,24,18,12)(8,10,20,22)(25,27,37,39)(26,32,38,44)(28,42,40,30)(29,47,41,35)(31,33,43,45)(34,48,46,36) );
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)], [(1,35,17,43,9,27),(2,40,18,48,10,32),(3,45,19,29,11,37),(4,26,20,34,12,42),(5,31,21,39,13,47),(6,36,22,44,14,28),(7,41,23,25,15,33),(8,46,24,30,16,38)], [(1,23,13,11),(2,4,14,16),(3,9,15,21),(5,19,17,7),(6,24,18,12),(8,10,20,22),(25,27,37,39),(26,32,38,44),(28,42,40,30),(29,47,41,35),(31,33,43,45),(34,48,46,36)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N | 24O | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 6 | 18 | 2 | 2 | 4 | 1 | 1 | 6 | 6 | 18 | 2 | 2 | 4 | 6 | 6 | 12 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S32 | C2×S32 | D12.C4 | C4×S32 | C24.64D6 |
| kernel | C24.64D6 | S3×C3⋊C8 | C12.29D6 | D6.Dic3 | S3×C24 | C3×C8⋊S3 | C24⋊S3 | D6.D6 | D6⋊S3 | C3⋊D12 | C32⋊2Q8 | S3×C8 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C32 | C3 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C24.64D6 ►in GL4(𝔽5) generated by
| 3 | 3 | 1 | 1 |
| 3 | 2 | 3 | 2 |
| 2 | 1 | 1 | 1 |
| 1 | 2 | 3 | 1 |
| 4 | 2 | 2 | 2 |
| 3 | 1 | 2 | 2 |
| 4 | 4 | 4 | 2 |
| 0 | 2 | 3 | 1 |
| 1 | 3 | 2 | 1 |
| 0 | 4 | 4 | 3 |
| 2 | 1 | 2 | 1 |
| 4 | 3 | 2 | 3 |
G:=sub<GL(4,GF(5))| [3,3,2,1,3,2,1,2,1,3,1,3,1,2,1,1],[4,3,4,0,2,1,4,2,2,2,4,3,2,2,2,1],[1,0,2,4,3,4,1,3,2,4,2,2,1,3,1,3] >;
C24.64D6 in GAP, Magma, Sage, TeX
C_{24}._{64}D_6 % in TeX
G:=Group("C24.64D6"); // GroupNames label
G:=SmallGroup(288,452);
// by ID
G=gap.SmallGroup(288,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^12*b^-1>;
// generators/relations