metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D24, C12.30D8, C12.26SD16, C42.261D6, (C4×C8)⋊5S3, (C4×C24)⋊5C2, C6.3(C2×D8), C2.5(C2×D24), C12⋊2Q8⋊2C2, (C2×C8).287D6, C2.D24⋊1C2, (C2×C4).80D12, C3⋊1(C4.4D8), C6.5(C2×SD16), C4.5(C24⋊C2), (C2×C12).377D4, C4⋊D12.2C2, C6.5(C4.4D4), (C2×D12).2C22, C22.90(C2×D12), C4⋊Dic3.5C22, C4.102(C4○D12), C12.218(C4○D4), (C2×C24).347C22, (C2×C12).723C23, (C4×C12).307C22, C2.10(C42⋊7S3), C2.8(C2×C24⋊C2), (C2×C6).106(C2×D4), (C2×C4).666(C22×S3), SmallGroup(192,253)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.5D24
G = < a,b,c | a4=b24=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 472 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C4.4D8, C2.D24, C4×C24, C12⋊2Q8, C4⋊D12, C4.5D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×D8, C2×SD16, C24⋊C2, D24, C2×D12, C4○D12, C4.4D8, C42⋊7S3, C2×C24⋊C2, C2×D24, C4.5D24
►On 96 points
(1 95 25 59)(2 96 26 60)(3 73 27 61)(4 74 28 62)(5 75 29 63)(6 76 30 64)(7 77 31 65)(8 78 32 66)(9 79 33 67)(10 80 34 68)(11 81 35 69)(12 82 36 70)(13 83 37 71)(14 84 38 72)(15 85 39 49)(16 86 40 50)(17 87 41 51)(18 88 42 52)(19 89 43 53)(20 90 44 54)(21 91 45 55)(22 92 46 56)(23 93 47 57)(24 94 48 58)
(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 48 25 24)(2 23 26 47)(3 46 27 22)(4 21 28 45)(5 44 29 20)(6 19 30 43)(7 42 31 18)(8 17 32 41)(9 40 33 16)(10 15 34 39)(11 38 35 14)(12 13 36 37)(49 68 85 80)(50 79 86 67)(51 66 87 78)(52 77 88 65)(53 64 89 76)(54 75 90 63)(55 62 91 74)(56 73 92 61)(57 60 93 96)(58 95 94 59)(69 72 81 84)(70 83 82 71)
G:=sub<Sym(96)| (1,95,25,59)(2,96,26,60)(3,73,27,61)(4,74,28,62)(5,75,29,63)(6,76,30,64)(7,77,31,65)(8,78,32,66)(9,79,33,67)(10,80,34,68)(11,81,35,69)(12,82,36,70)(13,83,37,71)(14,84,38,72)(15,85,39,49)(16,86,40,50)(17,87,41,51)(18,88,42,52)(19,89,43,53)(20,90,44,54)(21,91,45,55)(22,92,46,56)(23,93,47,57)(24,94,48,58), (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), (1,48,25,24)(2,23,26,47)(3,46,27,22)(4,21,28,45)(5,44,29,20)(6,19,30,43)(7,42,31,18)(8,17,32,41)(9,40,33,16)(10,15,34,39)(11,38,35,14)(12,13,36,37)(49,68,85,80)(50,79,86,67)(51,66,87,78)(52,77,88,65)(53,64,89,76)(54,75,90,63)(55,62,91,74)(56,73,92,61)(57,60,93,96)(58,95,94,59)(69,72,81,84)(70,83,82,71)>;
G:=Group( (1,95,25,59)(2,96,26,60)(3,73,27,61)(4,74,28,62)(5,75,29,63)(6,76,30,64)(7,77,31,65)(8,78,32,66)(9,79,33,67)(10,80,34,68)(11,81,35,69)(12,82,36,70)(13,83,37,71)(14,84,38,72)(15,85,39,49)(16,86,40,50)(17,87,41,51)(18,88,42,52)(19,89,43,53)(20,90,44,54)(21,91,45,55)(22,92,46,56)(23,93,47,57)(24,94,48,58), (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), (1,48,25,24)(2,23,26,47)(3,46,27,22)(4,21,28,45)(5,44,29,20)(6,19,30,43)(7,42,31,18)(8,17,32,41)(9,40,33,16)(10,15,34,39)(11,38,35,14)(12,13,36,37)(49,68,85,80)(50,79,86,67)(51,66,87,78)(52,77,88,65)(53,64,89,76)(54,75,90,63)(55,62,91,74)(56,73,92,61)(57,60,93,96)(58,95,94,59)(69,72,81,84)(70,83,82,71) );
G=PermutationGroup([[(1,95,25,59),(2,96,26,60),(3,73,27,61),(4,74,28,62),(5,75,29,63),(6,76,30,64),(7,77,31,65),(8,78,32,66),(9,79,33,67),(10,80,34,68),(11,81,35,69),(12,82,36,70),(13,83,37,71),(14,84,38,72),(15,85,39,49),(16,86,40,50),(17,87,41,51),(18,88,42,52),(19,89,43,53),(20,90,44,54),(21,91,45,55),(22,92,46,56),(23,93,47,57),(24,94,48,58)], [(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)], [(1,48,25,24),(2,23,26,47),(3,46,27,22),(4,21,28,45),(5,44,29,20),(6,19,30,43),(7,42,31,18),(8,17,32,41),(9,40,33,16),(10,15,34,39),(11,38,35,14),(12,13,36,37),(49,68,85,80),(50,79,86,67),(51,66,87,78),(52,77,88,65),(53,64,89,76),(54,75,90,63),(55,62,91,74),(56,73,92,61),(57,60,93,96),(58,95,94,59),(69,72,81,84),(70,83,82,71)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | ··· | 2 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D8 | SD16 | C4○D4 | D12 | C24⋊C2 | D24 | C4○D12 |
| kernel | C4.5D24 | C2.D24 | C4×C24 | C12⋊2Q8 | C4⋊D12 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C4.5D24 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 62 | 37 | 0 | 0 |
| 36 | 25 | 0 | 0 |
| 0 | 0 | 68 | 50 |
| 0 | 0 | 23 | 18 |
| 62 | 37 | 0 | 0 |
| 48 | 11 | 0 | 0 |
| 0 | 0 | 68 | 50 |
| 0 | 0 | 55 | 5 |
G:=sub<GL(4,GF(73))| [66,14,0,0,59,7,0,0,0,0,1,0,0,0,0,1],[62,36,0,0,37,25,0,0,0,0,68,23,0,0,50,18],[62,48,0,0,37,11,0,0,0,0,68,55,0,0,50,5] >;
C4.5D24 in GAP, Magma, Sage, TeX
C_4._5D_{24} % in TeX
G:=Group("C4.5D24"); // GroupNames label
G:=SmallGroup(192,253);
// by ID
G=gap.SmallGroup(192,253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations