metabelian, supersoluble, monomial
Aliases: C24.63D6, C8.24S32, (S3×C8)⋊6S3, C3⋊C8.29D6, D6.2(C4×S3), (S3×C24)⋊12C2, C3⋊1(C8○D12), (C4×S3).40D6, C32⋊1(C8○D4), D6⋊S3.3C4, C3⋊D12.3C4, Dic3.5(C4×S3), C32⋊2Q8.3C4, D6.Dic3⋊15C2, (C3×C24).45C22, D6.D6.5C2, C12.31D6⋊13C2, (S3×C12).38C22, (C3×C12).138C23, C12.137(C22×S3), C32⋊4C8.36C22, C2.6(C4×S32), C6.4(S3×C2×C4), C4.84(C2×S32), (C8×C3⋊S3)⋊11C2, (S3×C6).10(C2×C4), (C3×C3⋊C8).37C22, (C3×C6).4(C22×C4), (C4×C3⋊S3).86C22, C3⋊Dic3.31(C2×C4), (C3×Dic3).11(C2×C4), (C2×C3⋊S3).27(C2×C4), SmallGroup(288,451)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.63D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=cac-1=a17, cbc-1=a12b-1 >
Subgroups: 434 in 135 conjugacy classes, 50 normal (18 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, C4.Dic3, C2×C24, C4○D12, C3×C3⋊C8, C32⋊4C8, C3×C24, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, C8○D12, D6.Dic3, C12.31D6, S3×C24, C8×C3⋊S3, D6.D6, C24.63D6
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, C4×S32, C24.63D6
►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 32 17 40 9 48)(2 25 18 33 10 41)(3 42 19 26 11 34)(4 35 20 43 12 27)(5 28 21 36 13 44)(6 45 22 29 14 37)(7 38 23 46 15 30)(8 31 24 39 16 47)
(1 23 13 11)(2 16 14 4)(3 9 15 21)(5 19 17 7)(6 12 18 24)(8 22 20 10)(25 27 37 39)(26 44 38 32)(28 30 40 42)(29 47 41 35)(31 33 43 45)(34 36 46 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), (1,32,17,40,9,48)(2,25,18,33,10,41)(3,42,19,26,11,34)(4,35,20,43,12,27)(5,28,21,36,13,44)(6,45,22,29,14,37)(7,38,23,46,15,30)(8,31,24,39,16,47), (1,23,13,11)(2,16,14,4)(3,9,15,21)(5,19,17,7)(6,12,18,24)(8,22,20,10)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48)>;
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), (1,32,17,40,9,48)(2,25,18,33,10,41)(3,42,19,26,11,34)(4,35,20,43,12,27)(5,28,21,36,13,44)(6,45,22,29,14,37)(7,38,23,46,15,30)(8,31,24,39,16,47), (1,23,13,11)(2,16,14,4)(3,9,15,21)(5,19,17,7)(6,12,18,24)(8,22,20,10)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48) );
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)], [(1,32,17,40,9,48),(2,25,18,33,10,41),(3,42,19,26,11,34),(4,35,20,43,12,27),(5,28,21,36,13,44),(6,45,22,29,14,37),(7,38,23,46,15,30),(8,31,24,39,16,47)], [(1,23,13,11),(2,16,14,4),(3,9,15,21),(5,19,17,7),(6,12,18,24),(8,22,20,10),(25,27,37,39),(26,44,38,32),(28,30,40,42),(29,47,41,35),(31,33,43,45),(34,36,46,48)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 24M | ··· | 24T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 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 | 12 | 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 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S32 | C2×S32 | C4×S32 | C24.63D6 |
| kernel | C24.63D6 | D6.Dic3 | C12.31D6 | S3×C24 | C8×C3⋊S3 | D6.D6 | D6⋊S3 | C3⋊D12 | C32⋊2Q8 | S3×C8 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C32 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C24.63D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 46 | 46 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 65 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,0,46,0,0,0,0,27,46],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,46,65,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;
C24.63D6 in GAP, Magma, Sage, TeX
C_{24}._{63}D_6 % in TeX
G:=Group("C24.63D6"); // GroupNames label
G:=SmallGroup(288,451);
// by ID
G=gap.SmallGroup(288,451);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations