metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5D4, Dic3⋊1D8, C3⋊C8⋊14D4, (C6×D8)⋊5C2, (C2×D8)⋊3S3, C8⋊7(C3⋊D4), C3⋊3(C8⋊4D4), C6.44(C2×D8), C4.20(S3×D4), C2.27(S3×D8), (C2×D24)⋊18C2, C12⋊3D4⋊4C2, (C8×Dic3)⋊5C2, (C2×D4).60D6, C12.45(C2×D4), (C2×C8).236D6, C6.26(C4⋊1D4), (C2×C24).88C22, (C6×D4).78C22, C22.252(S3×D4), C2.17(C12⋊3D4), (C2×C12).428C23, (C2×Dic3).110D4, (C2×D12).114C22, (C4×Dic3).238C22, C4.4(C2×C3⋊D4), (C2×D4⋊S3)⋊17C2, (C2×C6).341(C2×D4), (C2×C3⋊C8).269C22, (C2×C4).518(C22×S3), SmallGroup(192,710)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊5D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a-1, cbc=b-1 >
Subgroups: 632 in 162 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×C8, C2×C8, D8, C2×D4, C2×D4, C3⋊C8, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4×C8, C4⋊1D4, C2×D8, C2×D8, D24, C2×C3⋊C8, C4×Dic3, D4⋊S3, C2×C24, C3×D8, C2×D12, C2×C3⋊D4, C6×D4, C8⋊4D4, C8×Dic3, C2×D24, C2×D4⋊S3, C12⋊3D4, C6×D8, C24⋊5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×D8, S3×D4, C2×C3⋊D4, C8⋊4D4, S3×D8, C12⋊3D4, C24⋊5D4
►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 46 52 85)(2 39 53 78)(3 32 54 95)(4 25 55 88)(5 42 56 81)(6 35 57 74)(7 28 58 91)(8 45 59 84)(9 38 60 77)(10 31 61 94)(11 48 62 87)(12 41 63 80)(13 34 64 73)(14 27 65 90)(15 44 66 83)(16 37 67 76)(17 30 68 93)(18 47 69 86)(19 40 70 79)(20 33 71 96)(21 26 72 89)(22 43 49 82)(23 36 50 75)(24 29 51 92)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 55)(50 54)(51 53)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)
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), (1,46,52,85)(2,39,53,78)(3,32,54,95)(4,25,55,88)(5,42,56,81)(6,35,57,74)(7,28,58,91)(8,45,59,84)(9,38,60,77)(10,31,61,94)(11,48,62,87)(12,41,63,80)(13,34,64,73)(14,27,65,90)(15,44,66,83)(16,37,67,76)(17,30,68,93)(18,47,69,86)(19,40,70,79)(20,33,71,96)(21,26,72,89)(22,43,49,82)(23,36,50,75)(24,29,51,92), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)>;
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), (1,46,52,85)(2,39,53,78)(3,32,54,95)(4,25,55,88)(5,42,56,81)(6,35,57,74)(7,28,58,91)(8,45,59,84)(9,38,60,77)(10,31,61,94)(11,48,62,87)(12,41,63,80)(13,34,64,73)(14,27,65,90)(15,44,66,83)(16,37,67,76)(17,30,68,93)(18,47,69,86)(19,40,70,79)(20,33,71,96)(21,26,72,89)(22,43,49,82)(23,36,50,75)(24,29,51,92), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65) );
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)], [(1,46,52,85),(2,39,53,78),(3,32,54,95),(4,25,55,88),(5,42,56,81),(6,35,57,74),(7,28,58,91),(8,45,59,84),(9,38,60,77),(10,31,61,94),(11,48,62,87),(12,41,63,80),(13,34,64,73),(14,27,65,90),(15,44,66,83),(16,37,67,76),(17,30,68,93),(18,47,69,86),(19,40,70,79),(20,33,71,96),(21,26,72,89),(22,43,49,82),(23,36,50,75),(24,29,51,92)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,82),(26,81),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83),(49,55),(50,54),(51,53),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | S3×D4 | S3×D4 | S3×D8 |
| kernel | C24⋊5D4 | C8×Dic3 | C2×D24 | C2×D4⋊S3 | C12⋊3D4 | C6×D8 | C2×D8 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×D4 | Dic3 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 1 | 1 | 4 |
Matrix representation of C24⋊5D4 ►in GL4(𝔽73) generated by
| 0 | 48 | 0 | 0 |
| 38 | 32 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 1 | 70 | 0 | 0 |
| 25 | 72 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
| 72 | 0 | 0 | 0 |
| 48 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [0,38,0,0,48,32,0,0,0,0,0,1,0,0,72,72],[1,25,0,0,70,72,0,0,0,0,60,30,0,0,43,13],[72,48,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C24⋊5D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_5D_4 % in TeX
G:=Group("C24:5D4"); // GroupNames label
G:=SmallGroup(192,710);
// by ID
G=gap.SmallGroup(192,710);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,422,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations