metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊1D8, C4⋊2D24, D12⋊8D4, C42.34D6, C4⋊C8⋊3S3, C6.7(C2×D8), (C2×D24)⋊7C2, C3⋊2(C4⋊D8), (C2×C8).21D6, C2.9(C2×D24), (C4×D12)⋊17C2, C4⋊D12⋊7C2, C4.130(S3×D4), C2.D24⋊7C2, (C2×C4).133D12, (C2×C12).122D4, C12.339(C2×D4), C6.38(C4⋊D4), C2.17(C8⋊D6), C6.14(C8⋊C22), (C4×C12).69C22, (C2×C24).22C22, C12.328(C4○D4), C2.11(C12⋊D4), (C2×C12).753C23, C4.44(Q8⋊3S3), (C2×D12).14C22, C22.116(C2×D12), C4⋊Dic3.273C22, (C3×C4⋊C8)⋊5C2, (C2×C6).136(C2×D4), (C2×C4).698(C22×S3), SmallGroup(192,402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D24
G = < a,b,c | a4=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 584 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, D24, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C2×D12, C2×D12, C4⋊D8, C2.D24, C3×C4⋊C8, C4×D12, C4⋊D12, C2×D24, C4⋊D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D8, C8⋊C22, D24, C2×D12, S3×D4, Q8⋊3S3, C4⋊D8, C12⋊D4, C2×D24, C8⋊D6, C4⋊D24
►On 96 points
(1 95 30 56)(2 57 31 96)(3 73 32 58)(4 59 33 74)(5 75 34 60)(6 61 35 76)(7 77 36 62)(8 63 37 78)(9 79 38 64)(10 65 39 80)(11 81 40 66)(12 67 41 82)(13 83 42 68)(14 69 43 84)(15 85 44 70)(16 71 45 86)(17 87 46 72)(18 49 47 88)(19 89 48 50)(20 51 25 90)(21 91 26 52)(22 53 27 92)(23 93 28 54)(24 55 29 94)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 96)(70 95)(71 94)(72 93)
G:=sub<Sym(96)| (1,95,30,56)(2,57,31,96)(3,73,32,58)(4,59,33,74)(5,75,34,60)(6,61,35,76)(7,77,36,62)(8,63,37,78)(9,79,38,64)(10,65,39,80)(11,81,40,66)(12,67,41,82)(13,83,42,68)(14,69,43,84)(15,85,44,70)(16,71,45,86)(17,87,46,72)(18,49,47,88)(19,89,48,50)(20,51,25,90)(21,91,26,52)(22,53,27,92)(23,93,28,54)(24,55,29,94), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,96)(70,95)(71,94)(72,93)>;
G:=Group( (1,95,30,56)(2,57,31,96)(3,73,32,58)(4,59,33,74)(5,75,34,60)(6,61,35,76)(7,77,36,62)(8,63,37,78)(9,79,38,64)(10,65,39,80)(11,81,40,66)(12,67,41,82)(13,83,42,68)(14,69,43,84)(15,85,44,70)(16,71,45,86)(17,87,46,72)(18,49,47,88)(19,89,48,50)(20,51,25,90)(21,91,26,52)(22,53,27,92)(23,93,28,54)(24,55,29,94), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,96)(70,95)(71,94)(72,93) );
G=PermutationGroup([[(1,95,30,56),(2,57,31,96),(3,73,32,58),(4,59,33,74),(5,75,34,60),(6,61,35,76),(7,77,36,62),(8,63,37,78),(9,79,38,64),(10,65,39,80),(11,81,40,66),(12,67,41,82),(13,83,42,68),(14,69,43,84),(15,85,44,70),(16,71,45,86),(17,87,46,72),(18,49,47,88),(19,89,48,50),(20,51,25,90),(21,91,26,52),(22,53,27,92),(23,93,28,54),(24,55,29,94)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,96),(70,95),(71,94),(72,93)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | C4○D4 | D12 | D24 | C8⋊C22 | S3×D4 | Q8⋊3S3 | C8⋊D6 |
| kernel | C4⋊D24 | C2.D24 | C3×C4⋊C8 | C4×D12 | C4⋊D12 | C2×D24 | C4⋊C8 | D12 | C2×C12 | C42 | C2×C8 | C12 | C12 | C2×C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of C4⋊D24 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 41 | 0 | 0 | 0 | 0 |
| 16 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 66 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,66,7,0,0,0,0,66,59,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1] >;
C4⋊D24 in GAP, Magma, Sage, TeX
C_4\rtimes D_{24} % in TeX
G:=Group("C4:D24"); // GroupNames label
G:=SmallGroup(192,402);
// by ID
G=gap.SmallGroup(192,402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations