metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊15D4, Dic3⋊1SD16, C3⋊C8⋊15D4, C8⋊8(C3⋊D4), C3⋊4(C8⋊5D4), C4.25(S3×D4), (C6×SD16)⋊9C2, (C2×D4).76D6, C12.50(C2×D4), (C2×C8).264D6, (C2×Q8).81D6, (C8×Dic3)⋊11C2, (C2×SD16)⋊15S3, C12⋊3D4.6C2, C6.48(C2×SD16), C2.31(S3×SD16), Dic3⋊Q8⋊5C2, C6.31(C4⋊1D4), C22.272(S3×D4), (C6×Q8).81C22, C2.22(C12⋊3D4), (C2×C24).165C22, (C2×C12).452C23, (C2×Dic3).115D4, (C6×D4).101C22, (C2×D12).122C22, (C2×Dic6).129C22, (C4×Dic3).242C22, C4.9(C2×C3⋊D4), (C2×C24⋊C2)⋊30C2, (C2×D4.S3)⋊21C2, (C2×C6).364(C2×D4), (C2×Q8⋊2S3)⋊18C2, (C2×C3⋊C8).275C22, (C2×C4).541(C22×S3), SmallGroup(192,734)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊15D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a11, cbc=b-1 >
Subgroups: 472 in 142 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D4.S3, Q8⋊2S3, C2×C24, C3×SD16, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C6×Q8, C8⋊5D4, C8×Dic3, C2×C24⋊C2, C2×D4.S3, C12⋊3D4, C2×Q8⋊2S3, Dic3⋊Q8, C6×SD16, C24⋊15D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×SD16, S3×D4, C2×C3⋊D4, C8⋊5D4, S3×SD16, C12⋊3D4, C24⋊15D4
►On 96 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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 76 35 51)(2 93 36 68)(3 86 37 61)(4 79 38 54)(5 96 39 71)(6 89 40 64)(7 82 41 57)(8 75 42 50)(9 92 43 67)(10 85 44 60)(11 78 45 53)(12 95 46 70)(13 88 47 63)(14 81 48 56)(15 74 25 49)(16 91 26 66)(17 84 27 59)(18 77 28 52)(19 94 29 69)(20 87 30 62)(21 80 31 55)(22 73 32 72)(23 90 33 65)(24 83 34 58)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 45)(26 32)(27 43)(28 30)(29 41)(31 39)(33 37)(34 48)(36 46)(38 44)(40 42)(49 78)(50 89)(51 76)(52 87)(53 74)(54 85)(55 96)(56 83)(57 94)(58 81)(59 92)(60 79)(61 90)(62 77)(63 88)(64 75)(65 86)(66 73)(67 84)(68 95)(69 82)(70 93)(71 80)(72 91)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,76,35,51)(2,93,36,68)(3,86,37,61)(4,79,38,54)(5,96,39,71)(6,89,40,64)(7,82,41,57)(8,75,42,50)(9,92,43,67)(10,85,44,60)(11,78,45,53)(12,95,46,70)(13,88,47,63)(14,81,48,56)(15,74,25,49)(16,91,26,66)(17,84,27,59)(18,77,28,52)(19,94,29,69)(20,87,30,62)(21,80,31,55)(22,73,32,72)(23,90,33,65)(24,83,34,58), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,45)(26,32)(27,43)(28,30)(29,41)(31,39)(33,37)(34,48)(36,46)(38,44)(40,42)(49,78)(50,89)(51,76)(52,87)(53,74)(54,85)(55,96)(56,83)(57,94)(58,81)(59,92)(60,79)(61,90)(62,77)(63,88)(64,75)(65,86)(66,73)(67,84)(68,95)(69,82)(70,93)(71,80)(72,91)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,76,35,51)(2,93,36,68)(3,86,37,61)(4,79,38,54)(5,96,39,71)(6,89,40,64)(7,82,41,57)(8,75,42,50)(9,92,43,67)(10,85,44,60)(11,78,45,53)(12,95,46,70)(13,88,47,63)(14,81,48,56)(15,74,25,49)(16,91,26,66)(17,84,27,59)(18,77,28,52)(19,94,29,69)(20,87,30,62)(21,80,31,55)(22,73,32,72)(23,90,33,65)(24,83,34,58), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,45)(26,32)(27,43)(28,30)(29,41)(31,39)(33,37)(34,48)(36,46)(38,44)(40,42)(49,78)(50,89)(51,76)(52,87)(53,74)(54,85)(55,96)(56,83)(57,94)(58,81)(59,92)(60,79)(61,90)(62,77)(63,88)(64,75)(65,86)(66,73)(67,84)(68,95)(69,82)(70,93)(71,80)(72,91) );
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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,76,35,51),(2,93,36,68),(3,86,37,61),(4,79,38,54),(5,96,39,71),(6,89,40,64),(7,82,41,57),(8,75,42,50),(9,92,43,67),(10,85,44,60),(11,78,45,53),(12,95,46,70),(13,88,47,63),(14,81,48,56),(15,74,25,49),(16,91,26,66),(17,84,27,59),(18,77,28,52),(19,94,29,69),(20,87,30,62),(21,80,31,55),(22,73,32,72),(23,90,33,65),(24,83,34,58)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,45),(26,32),(27,43),(28,30),(29,41),(31,39),(33,37),(34,48),(36,46),(38,44),(40,42),(49,78),(50,89),(51,76),(52,87),(53,74),(54,85),(55,96),(56,83),(57,94),(58,81),(59,92),(60,79),(61,90),(62,77),(63,88),(64,75),(65,86),(66,73),(67,84),(68,95),(69,82),(70,93),(71,80),(72,91)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | C3⋊D4 | S3×D4 | S3×D4 | S3×SD16 |
| kernel | C24⋊15D4 | C8×Dic3 | C2×C24⋊C2 | C2×D4.S3 | C12⋊3D4 | C2×Q8⋊2S3 | Dic3⋊Q8 | C6×SD16 | C2×SD16 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×D4 | C2×Q8 | Dic3 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 1 | 1 | 4 |
Matrix representation of C24⋊15D4 ►in GL6(𝔽73)
| 6 | 67 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 67 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [6,6,0,0,0,0,67,6,0,0,0,0,0,0,6,6,0,0,0,0,67,6,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C24⋊15D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{15}D_4 % in TeX
G:=Group("C24:15D4"); // GroupNames label
G:=SmallGroup(192,734);
// by ID
G=gap.SmallGroup(192,734);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,555,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations