metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊9C4, C8⋊4(C4×S3), C24⋊8(C2×C4), C4.Q8⋊5S3, D12⋊6(C2×C4), C24⋊C4⋊3C2, (C2×C8).63D6, C6.51(C4×D4), C3⋊3(D8⋊C4), C4⋊C4.163D6, Dic3⋊5D4⋊6C2, (C2×D24).13C2, C6.D8⋊18C2, C2.6(Q8⋊3D6), C22.86(S3×D4), C12.34(C4○D4), C6.72(C8⋊C22), C12.45(C22×C4), C4.6(Q8⋊3S3), (C2×C24).112C22, (C2×C12).285C23, (C2×Dic3).165D4, (C2×D12).77C22, C2.11(Dic3⋊5D4), (C4×Dic3).31C22, C4.42(S3×C2×C4), (C3×C4.Q8)⋊5C2, (C2×C6).290(C2×D4), (C2×C3⋊C8).62C22, (C3×C4⋊C4).78C22, (C2×C4).388(C22×S3), SmallGroup(192,428)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊9C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a18b >
Subgroups: 448 in 132 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, D24, C2×C3⋊C8, C4×Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, D8⋊C4, C6.D8, C24⋊C4, C3×C4.Q8, Dic3⋊5D4, C2×D24, D24⋊9C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8⋊C22, S3×C2×C4, S3×D4, Q8⋊3S3, D8⋊C4, Dic3⋊5D4, Q8⋊3D6, D24⋊9C4
►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 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 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(68 72)(69 71)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 39 63 87)(2 34 64 82)(3 29 65 77)(4 48 66 96)(5 43 67 91)(6 38 68 86)(7 33 69 81)(8 28 70 76)(9 47 71 95)(10 42 72 90)(11 37 49 85)(12 32 50 80)(13 27 51 75)(14 46 52 94)(15 41 53 89)(16 36 54 84)(17 31 55 79)(18 26 56 74)(19 45 57 93)(20 40 58 88)(21 35 59 83)(22 30 60 78)(23 25 61 73)(24 44 62 92)
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), (1,15)(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)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,72)(69,71)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,39,63,87)(2,34,64,82)(3,29,65,77)(4,48,66,96)(5,43,67,91)(6,38,68,86)(7,33,69,81)(8,28,70,76)(9,47,71,95)(10,42,72,90)(11,37,49,85)(12,32,50,80)(13,27,51,75)(14,46,52,94)(15,41,53,89)(16,36,54,84)(17,31,55,79)(18,26,56,74)(19,45,57,93)(20,40,58,88)(21,35,59,83)(22,30,60,78)(23,25,61,73)(24,44,62,92)>;
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), (1,15)(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)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,72)(69,71)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86), (1,39,63,87)(2,34,64,82)(3,29,65,77)(4,48,66,96)(5,43,67,91)(6,38,68,86)(7,33,69,81)(8,28,70,76)(9,47,71,95)(10,42,72,90)(11,37,49,85)(12,32,50,80)(13,27,51,75)(14,46,52,94)(15,41,53,89)(16,36,54,84)(17,31,55,79)(18,26,56,74)(19,45,57,93)(20,40,58,88)(21,35,59,83)(22,30,60,78)(23,25,61,73)(24,44,62,92) );
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)], [(1,15),(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),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(68,72),(69,71),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)], [(1,39,63,87),(2,34,64,82),(3,29,65,77),(4,48,66,96),(5,43,67,91),(6,38,68,86),(7,33,69,81),(8,28,70,76),(9,47,71,95),(10,42,72,90),(11,37,49,85),(12,32,50,80),(13,27,51,75),(14,46,52,94),(15,41,53,89),(16,36,54,84),(17,31,55,79),(18,26,56,74),(19,45,57,93),(20,40,58,88),(21,35,59,83),(22,30,60,78),(23,25,61,73),(24,44,62,92)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C8⋊C22 | Q8⋊3S3 | S3×D4 | Q8⋊3D6 |
| kernel | D24⋊9C4 | C6.D8 | C24⋊C4 | C3×C4.Q8 | Dic3⋊5D4 | C2×D24 | D24 | C4.Q8 | C2×Dic3 | C4⋊C4 | C2×C8 | C12 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 4 |
Matrix representation of D24⋊9C4 ►in GL8(𝔽73)
| 48 | 48 | 54 | 35 | 0 | 0 | 0 | 0 |
| 27 | 19 | 19 | 54 | 0 | 0 | 0 | 0 |
| 56 | 38 | 54 | 25 | 0 | 0 | 0 | 0 |
| 36 | 56 | 46 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 68 | 0 |
| 0 | 0 | 0 | 0 | 29 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 29 | 29 | 68 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 55 | 0 | 1 | 0 | 0 | 0 | 0 |
| 9 | 55 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 50 | 48 |
| 0 | 0 | 0 | 0 | 1 | 0 | 48 | 48 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 27 | 27 | 71 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 43 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 30 | 13 |
| 0 | 0 | 0 | 0 | 56 | 0 | 60 | 43 |
G:=sub<GL(8,GF(73))| [48,27,56,36,0,0,0,0,48,19,38,56,0,0,0,0,54,19,54,46,0,0,0,0,35,54,25,25,0,0,0,0,0,0,0,0,0,0,29,29,0,0,0,0,0,0,0,29,0,0,0,0,0,68,5,68,0,0,0,0,5,0,5,10],[1,1,9,9,0,0,0,0,0,72,55,55,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,50,48,1,1,0,0,0,0,48,48,0,72],[0,27,0,72,0,0,0,0,0,27,0,0,0,0,0,0,72,71,46,46,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,30,13,0,56,0,0,0,0,60,43,17,0,0,0,0,0,0,60,30,60,0,0,0,0,13,0,13,43] >;
D24⋊9C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes_9C_4 % in TeX
G:=Group("D24:9C4"); // GroupNames label
G:=SmallGroup(192,428);
// by ID
G=gap.SmallGroup(192,428);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,219,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations