metabelian, supersoluble, monomial
Aliases: C24⋊9D6, Dic6⋊1D6, D12.1D6, C8⋊8S32, C24⋊C2⋊5S3, C3⋊S3⋊3SD16, C3⋊2(S3×SD16), C6.28(S3×D4), C32⋊5(C2×SD16), (C3×C24)⋊15C22, D6⋊D6.3C2, C3⋊Dic3.39D4, Dic3.D6⋊1C2, Dic6⋊S3⋊1C2, C2.5(D6⋊D6), C12.67(C22×S3), (C3×C12).44C23, (C3×Dic6)⋊3C22, (C3×D12).3C22, C32⋊4C8⋊16C22, (C8×C3⋊S3)⋊6C2, C4.65(C2×S32), (C2×C3⋊S3).40D4, (C3×C6).28(C2×D4), (C3×C24⋊C2)⋊10C2, (C4×C3⋊S3).65C22, SmallGroup(288,444)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊9D6
G = < a,b,c | a24=b6=c2=1, bab-1=a11, cac=a17, cbc=b-1 >
Subgroups: 690 in 147 conjugacy classes, 40 normal (16 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, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C32⋊4C8, C3×C24, C6.D6, D6⋊S3, C32⋊2Q8, C3×Dic6, C3×D12, C4×C3⋊S3, C2×S32, S3×SD16, Dic6⋊S3, C3×C24⋊C2, C8×C3⋊S3, Dic3.D6, D6⋊D6, C24⋊9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C22×S3, C2×SD16, S32, S3×D4, C2×S32, S3×SD16, D6⋊D6, C24⋊9D6
►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 44 17 28 9 36)(2 31 18 39 10 47)(3 42 19 26 11 34)(4 29 20 37 12 45)(5 40 21 48 13 32)(6 27 22 35 14 43)(7 38 23 46 15 30)(8 25 24 33 16 41)
(1 17)(2 10)(4 20)(5 13)(7 23)(8 16)(11 19)(14 22)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(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), (1,44,17,28,9,36)(2,31,18,39,10,47)(3,42,19,26,11,34)(4,29,20,37,12,45)(5,40,21,48,13,32)(6,27,22,35,14,43)(7,38,23,46,15,30)(8,25,24,33,16,41), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(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), (1,44,17,28,9,36)(2,31,18,39,10,47)(3,42,19,26,11,34)(4,29,20,37,12,45)(5,40,21,48,13,32)(6,27,22,35,14,43)(7,38,23,46,15,30)(8,25,24,33,16,41), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(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)], [(1,44,17,28,9,36),(2,31,18,39,10,47),(3,42,19,26,11,34),(4,29,20,37,12,45),(5,40,21,48,13,32),(6,27,22,35,14,43),(7,38,23,46,15,30),(8,25,24,33,16,41)], [(1,17),(2,10),(4,20),(5,13),(7,23),(8,16),(11,19),(14,22),(25,33),(27,43),(28,36),(30,46),(31,39),(34,42),(37,45),(40,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 12 | 12 | 2 | 2 | 4 | 2 | 12 | 12 | 18 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | S32 | S3×D4 | C2×S32 | S3×SD16 | D6⋊D6 | C24⋊9D6 |
| kernel | C24⋊9D6 | Dic6⋊S3 | C3×C24⋊C2 | C8×C3⋊S3 | Dic3.D6 | D6⋊D6 | C24⋊C2 | C3⋊Dic3 | C2×C3⋊S3 | C24 | Dic6 | D12 | C3⋊S3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 4 |
Matrix representation of C24⋊9D6 ►in GL6(𝔽73)
| 63 | 55 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 56 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [63,0,0,0,0,0,55,22,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[5,56,0,0,0,0,10,68,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C24⋊9D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes_9D_6 % in TeX
G:=Group("C24:9D6"); // GroupNames label
G:=SmallGroup(288,444);
// by ID
G=gap.SmallGroup(288,444);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,58,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^11,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations