metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊3C4, C42.17D6, C8⋊2(C4×S3), C24⋊2(C2×C4), C8⋊C4⋊2S3, (C4×D12)⋊2C2, D12⋊9(C2×C4), C8⋊Dic3⋊2C2, C6.12(C4×D4), (C2×C8).54D6, C3⋊1(D8⋊C4), (C2×D24).7C2, C2.15(C4×D12), C2.D24⋊37C2, (C2×C12).237D4, (C2×C4).115D12, C2.2(C8⋊D6), C6.3(C8⋊C22), (C2×C24).55C22, (C4×C12).15C22, C22.31(C2×D12), C4.107(C4○D12), C12.223(C4○D4), C12.106(C22×C4), (C2×C12).732C23, (C2×D12).190C22, C4⋊Dic3.266C22, C4.64(S3×C2×C4), (C3×C8⋊C4)⋊3C2, (C2×C6).115(C2×D4), (C2×C4).676(C22×S3), SmallGroup(192,270)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a12b >
Subgroups: 456 in 132 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, D24, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, D8⋊C4, C8⋊Dic3, C2.D24, C3×C8⋊C4, C4×D12, C2×D24, D24⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C8⋊C22, S3×C2×C4, C2×D12, C4○D12, D8⋊C4, C4×D12, C8⋊D6, D24⋊C4
►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 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 96)(23 95)(24 94)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)
(1 64 79 46)(2 53 80 35)(3 66 81 48)(4 55 82 37)(5 68 83 26)(6 57 84 39)(7 70 85 28)(8 59 86 41)(9 72 87 30)(10 61 88 43)(11 50 89 32)(12 63 90 45)(13 52 91 34)(14 65 92 47)(15 54 93 36)(16 67 94 25)(17 56 95 38)(18 69 96 27)(19 58 73 40)(20 71 74 29)(21 60 75 42)(22 49 76 31)(23 62 77 44)(24 51 78 33)
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), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,96)(23,95)(24,94)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64), (1,64,79,46)(2,53,80,35)(3,66,81,48)(4,55,82,37)(5,68,83,26)(6,57,84,39)(7,70,85,28)(8,59,86,41)(9,72,87,30)(10,61,88,43)(11,50,89,32)(12,63,90,45)(13,52,91,34)(14,65,92,47)(15,54,93,36)(16,67,94,25)(17,56,95,38)(18,69,96,27)(19,58,73,40)(20,71,74,29)(21,60,75,42)(22,49,76,31)(23,62,77,44)(24,51,78,33)>;
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), (1,93)(2,92)(3,91)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,84)(11,83)(12,82)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,96)(23,95)(24,94)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64), (1,64,79,46)(2,53,80,35)(3,66,81,48)(4,55,82,37)(5,68,83,26)(6,57,84,39)(7,70,85,28)(8,59,86,41)(9,72,87,30)(10,61,88,43)(11,50,89,32)(12,63,90,45)(13,52,91,34)(14,65,92,47)(15,54,93,36)(16,67,94,25)(17,56,95,38)(18,69,96,27)(19,58,73,40)(20,71,74,29)(21,60,75,42)(22,49,76,31)(23,62,77,44)(24,51,78,33) );
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)], [(1,93),(2,92),(3,91),(4,90),(5,89),(6,88),(7,87),(8,86),(9,85),(10,84),(11,83),(12,82),(13,81),(14,80),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,73),(22,96),(23,95),(24,94),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64)], [(1,64,79,46),(2,53,80,35),(3,66,81,48),(4,55,82,37),(5,68,83,26),(6,57,84,39),(7,70,85,28),(8,59,86,41),(9,72,87,30),(10,61,88,43),(11,50,89,32),(12,63,90,45),(13,52,91,34),(14,65,92,47),(15,54,93,36),(16,67,94,25),(17,56,95,38),(18,69,96,27),(19,58,73,40),(20,71,74,29),(21,60,75,42),(22,49,76,31),(23,62,77,44),(24,51,78,33)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 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 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | C8⋊C22 | C8⋊D6 |
| kernel | D24⋊C4 | C8⋊Dic3 | C2.D24 | C3×C8⋊C4 | C4×D12 | C2×D24 | D24 | C8⋊C4 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of D24⋊C4 ►in GL6(𝔽73)
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 68 | 28 | 16 |
| 0 | 0 | 5 | 54 | 57 | 12 |
| 0 | 0 | 54 | 19 | 24 | 5 |
| 0 | 0 | 54 | 35 | 68 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 |
| 0 | 0 | 36 | 18 | 59 | 66 |
| 0 | 0 | 55 | 37 | 7 | 14 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 23 | 71 | 0 |
| 0 | 0 | 50 | 38 | 0 | 71 |
| 0 | 0 | 26 | 7 | 12 | 50 |
| 0 | 0 | 66 | 19 | 23 | 35 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,49,5,54,54,0,0,68,54,19,35,0,0,28,57,24,68,0,0,16,12,5,19],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,14,66,36,55,0,0,7,59,18,37,0,0,0,0,59,7,0,0,0,0,66,14],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,61,50,26,66,0,0,23,38,7,19,0,0,71,0,12,23,0,0,0,71,50,35] >;
D24⋊C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes C_4 % in TeX
G:=Group("D24:C4"); // GroupNames label
G:=SmallGroup(192,270);
// by ID
G=gap.SmallGroup(192,270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,387,58,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations