metabelian, supersoluble, monomial
Aliases: C24⋊3D6, C12.32D12, C62.70D4, C32⋊5D8⋊3C2, C24⋊2S3⋊3C2, C3⋊3(C8⋊D6), (C3×C24)⋊5C22, (C2×C6).17D12, (C3×C12).95D4, C6.60(C2×D12), (C2×C12).147D6, (C3×M4(2))⋊1S3, C12.59D6⋊4C2, M4(2)⋊1(C3⋊S3), C32⋊17(C8⋊C22), C4.14(C12⋊S3), C12⋊S3⋊15C22, (C3×C12).155C23, C12.193(C22×S3), (C6×C12).138C22, (C32×M4(2))⋊3C2, C32⋊4Q8⋊14C22, C22.5(C12⋊S3), C8⋊1(C2×C3⋊S3), (C3×C6).200(C2×D4), (C2×C12⋊S3)⋊13C2, C4.30(C22×C3⋊S3), C2.15(C2×C12⋊S3), (C2×C4).15(C2×C3⋊S3), SmallGroup(288,765)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊3D6
G = < a,b,c | a24=b6=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >
Subgroups: 1100 in 204 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3⋊S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C22×S3, C8⋊C22, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C12⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C22×C3⋊S3, C8⋊D6, C24⋊2S3, C32⋊5D8, C32×M4(2), C2×C12⋊S3, C12.59D6, C24⋊3D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C8⋊C22, C2×C3⋊S3, C2×D12, C12⋊S3, C22×C3⋊S3, C8⋊D6, C2×C12⋊S3, C24⋊3D6
►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 62 34 13 50 46)(2 51 35)(3 64 36 15 52 48)(4 53 37)(5 66 38 17 54 26)(6 55 39)(7 68 40 19 56 28)(8 57 41)(9 70 42 21 58 30)(10 59 43)(11 72 44 23 60 32)(12 61 45)(14 63 47)(16 65 25)(18 67 27)(20 69 29)(22 71 31)(24 49 33)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 48)(24 47)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 72)(65 71)(66 70)(67 69)
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), (1,62,34,13,50,46)(2,51,35)(3,64,36,15,52,48)(4,53,37)(5,66,38,17,54,26)(6,55,39)(7,68,40,19,56,28)(8,57,41)(9,70,42,21,58,30)(10,59,43)(11,72,44,23,60,32)(12,61,45)(14,63,47)(16,65,25)(18,67,27)(20,69,29)(22,71,31)(24,49,33), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,48)(24,47)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69)>;
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), (1,62,34,13,50,46)(2,51,35)(3,64,36,15,52,48)(4,53,37)(5,66,38,17,54,26)(6,55,39)(7,68,40,19,56,28)(8,57,41)(9,70,42,21,58,30)(10,59,43)(11,72,44,23,60,32)(12,61,45)(14,63,47)(16,65,25)(18,67,27)(20,69,29)(22,71,31)(24,49,33), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,48)(24,47)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69) );
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)], [(1,62,34,13,50,46),(2,51,35),(3,64,36,15,52,48),(4,53,37),(5,66,38,17,54,26),(6,55,39),(7,68,40,19,56,28),(8,57,41),(9,70,42,21,58,30),(10,59,43),(11,72,44,23,60,32),(12,61,45),(14,63,47),(16,65,25),(18,67,27),(20,69,29),(22,71,31),(24,49,33)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,48),(24,47),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(64,72),(65,71),(66,70),(67,69)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C8⋊C22 | C8⋊D6 |
| kernel | C24⋊3D6 | C24⋊2S3 | C32⋊5D8 | C32×M4(2) | C2×C12⋊S3 | C12.59D6 | C3×M4(2) | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 1 | 8 |
Matrix representation of C24⋊3D6 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 66 |
| 0 | 0 | 0 | 0 | 59 | 7 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,66,59,0,0,0,0,66,7] >;
C24⋊3D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes_3D_6 % in TeX
G:=Group("C24:3D6"); // GroupNames label
G:=SmallGroup(288,765);
// by ID
G=gap.SmallGroup(288,765);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations