metabelian, supersoluble, monomial
Aliases: C24⋊8D6, (C3×D4)⋊4D6, (C3×D8)⋊4S3, D8⋊2(C3⋊S3), C24⋊2S3⋊5C2, C24⋊S3⋊5C2, C3⋊5(D8⋊S3), C6.119(S3×D4), (C32×D8)⋊7C2, C32⋊7D8⋊6C2, (C3×C24)⋊12C22, C3⋊Dic3.65D4, C12.D6⋊3C2, C32⋊9SD16⋊5C2, C32⋊19(C8⋊C22), (C3×C12).92C23, C12.88(C22×S3), (D4×C32)⋊8C22, C32⋊4Q8⋊7C22, C32⋊4C8⋊10C22, C12⋊S3.16C22, C8⋊2(C2×C3⋊S3), D4⋊2(C2×C3⋊S3), (D4×C3⋊S3)⋊4C2, C2.16(D4×C3⋊S3), (C2×C3⋊S3).65D4, C4.2(C22×C3⋊S3), (C3×C6).240(C2×D4), (C4×C3⋊S3).23C22, SmallGroup(288,768)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊8D6
G = < a,b,c | a24=b6=c2=1, bab-1=a7, cac=a11, cbc=b-1 >
Subgroups: 972 in 204 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C8⋊C22, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C32⋊4C8, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C22×C3⋊S3, D8⋊S3, C24⋊S3, C24⋊2S3, C32⋊7D8, C32⋊9SD16, C32×D8, D4×C3⋊S3, C12.D6, C24⋊8D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C8⋊C22, C2×C3⋊S3, S3×D4, C22×C3⋊S3, D8⋊S3, D4×C3⋊S3, C24⋊8D6
►On 72 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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 71 33)(2 54 34 8 72 40)(3 61 35 15 49 47)(4 68 36 22 50 30)(5 51 37)(6 58 38 12 52 44)(7 65 39 19 53 27)(9 55 41)(10 62 42 16 56 48)(11 69 43 23 57 31)(13 59 45)(14 66 46 20 60 28)(17 63 25)(18 70 26 24 64 32)(21 67 29)
(1 45)(2 32)(3 43)(4 30)(5 41)(6 28)(7 39)(8 26)(9 37)(10 48)(11 35)(12 46)(13 33)(14 44)(15 31)(16 42)(17 29)(18 40)(19 27)(20 38)(21 25)(22 36)(23 47)(24 34)(49 57)(50 68)(51 55)(52 66)(54 64)(56 62)(58 60)(59 71)(61 69)(63 67)(70 72)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,71,33)(2,54,34,8,72,40)(3,61,35,15,49,47)(4,68,36,22,50,30)(5,51,37)(6,58,38,12,52,44)(7,65,39,19,53,27)(9,55,41)(10,62,42,16,56,48)(11,69,43,23,57,31)(13,59,45)(14,66,46,20,60,28)(17,63,25)(18,70,26,24,64,32)(21,67,29), (1,45)(2,32)(3,43)(4,30)(5,41)(6,28)(7,39)(8,26)(9,37)(10,48)(11,35)(12,46)(13,33)(14,44)(15,31)(16,42)(17,29)(18,40)(19,27)(20,38)(21,25)(22,36)(23,47)(24,34)(49,57)(50,68)(51,55)(52,66)(54,64)(56,62)(58,60)(59,71)(61,69)(63,67)(70,72)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,71,33)(2,54,34,8,72,40)(3,61,35,15,49,47)(4,68,36,22,50,30)(5,51,37)(6,58,38,12,52,44)(7,65,39,19,53,27)(9,55,41)(10,62,42,16,56,48)(11,69,43,23,57,31)(13,59,45)(14,66,46,20,60,28)(17,63,25)(18,70,26,24,64,32)(21,67,29), (1,45)(2,32)(3,43)(4,30)(5,41)(6,28)(7,39)(8,26)(9,37)(10,48)(11,35)(12,46)(13,33)(14,44)(15,31)(16,42)(17,29)(18,40)(19,27)(20,38)(21,25)(22,36)(23,47)(24,34)(49,57)(50,68)(51,55)(52,66)(54,64)(56,62)(58,60)(59,71)(61,69)(63,67)(70,72) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,71,33),(2,54,34,8,72,40),(3,61,35,15,49,47),(4,68,36,22,50,30),(5,51,37),(6,58,38,12,52,44),(7,65,39,19,53,27),(9,55,41),(10,62,42,16,56,48),(11,69,43,23,57,31),(13,59,45),(14,66,46,20,60,28),(17,63,25),(18,70,26,24,64,32),(21,67,29)], [(1,45),(2,32),(3,43),(4,30),(5,41),(6,28),(7,39),(8,26),(9,37),(10,48),(11,35),(12,46),(13,33),(14,44),(15,31),(16,42),(17,29),(18,40),(19,27),(20,38),(21,25),(22,36),(23,47),(24,34),(49,57),(50,68),(51,55),(52,66),(54,64),(56,62),(58,60),(59,71),(61,69),(63,67),(70,72)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 4 | 18 | 36 | 2 | 2 | 2 | 2 | 2 | 18 | 36 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 36 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C8⋊C22 | S3×D4 | D8⋊S3 |
| kernel | C24⋊8D6 | C24⋊S3 | C24⋊2S3 | C32⋊7D8 | C32⋊9SD16 | C32×D8 | D4×C3⋊S3 | C12.D6 | C3×D8 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×D4 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 1 | 4 | 8 |
Matrix representation of C24⋊8D6 ►in GL6(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 62 |
| 0 | 0 | 0 | 0 | 11 | 62 |
| 0 | 0 | 11 | 42 | 51 | 62 |
| 0 | 0 | 31 | 42 | 11 | 62 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 1 | 1 |
| 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 72 | 72 |
| 0 | 0 | 0 | 72 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,31,0,0,0,0,42,42,0,0,51,11,51,11,0,0,62,62,62,62],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,1,0,0,0,1,72,1,72,0,0,0,0,72,0,0,0,0,0,72,1] >;
C24⋊8D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes_8D_6 % in TeX
G:=Group("C24:8D6"); // GroupNames label
G:=SmallGroup(288,768);
// by ID
G=gap.SmallGroup(288,768);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^7,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations