metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊21D4, Dic3⋊2M4(2), D6⋊C8⋊39C2, C8⋊9(C3⋊D4), C3⋊5(C8⋊6D4), C6.80(C4×D4), D6⋊C4.17C4, (C2×C8).276D6, Dic3⋊C8⋊41C2, (C8×Dic3)⋊31C2, C6.33(C8○D4), C12.443(C2×D4), (C2×M4(2))⋊8S3, C23.25(C4×S3), (C6×M4(2))⋊12C2, Dic3⋊C4.17C4, (C22×C4).153D6, C2.22(S3×M4(2)), C6.33(C2×M4(2)), C4.138(C4○D12), C12.254(C4○D4), C12.55D4⋊30C2, C2.18(D12.C4), (C2×C12).868C23, (C2×C24).318C22, C6.D4.13C4, (C22×C12).375C22, (C4×Dic3).286C22, (C2×C4).51(C4×S3), C2.25(C4×C3⋊D4), (C2×C8⋊S3)⋊25C2, (C4×C3⋊D4).17C2, (C2×C3⋊D4).11C4, C4.134(C2×C3⋊D4), C22.147(S3×C2×C4), (C2×C12).188(C2×C4), (C2×C3⋊C8).324C22, (S3×C2×C4).185C22, (C22×C6).68(C2×C4), (C22×S3).25(C2×C4), (C2×C4).810(C22×S3), (C2×C6).138(C22×C4), (C2×Dic3).68(C2×C4), SmallGroup(192,687)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊21D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a5, cbc=b-1 >
Subgroups: 280 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, C3×M4(2), S3×C2×C4, C2×C3⋊D4, C22×C12, C8⋊6D4, C8×Dic3, Dic3⋊C8, D6⋊C8, C12.55D4, C2×C8⋊S3, C4×C3⋊D4, C6×M4(2), C24⋊21D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C8⋊6D4, S3×M4(2), D12.C4, C4×C3⋊D4, C24⋊21D4
►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 35 62 96)(2 28 63 89)(3 45 64 82)(4 38 65 75)(5 31 66 92)(6 48 67 85)(7 41 68 78)(8 34 69 95)(9 27 70 88)(10 44 71 81)(11 37 72 74)(12 30 49 91)(13 47 50 84)(14 40 51 77)(15 33 52 94)(16 26 53 87)(17 43 54 80)(18 36 55 73)(19 29 56 90)(20 46 57 83)(21 39 58 76)(22 32 59 93)(23 25 60 86)(24 42 61 79)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 94)(26 75)(27 80)(28 85)(29 90)(30 95)(31 76)(32 81)(33 86)(34 91)(35 96)(36 77)(37 82)(38 87)(39 92)(40 73)(41 78)(42 83)(43 88)(44 93)(45 74)(46 79)(47 84)(48 89)(49 69)(51 55)(52 60)(53 65)(54 70)(57 61)(58 66)(59 71)(63 67)(64 72)
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), (1,35,62,96)(2,28,63,89)(3,45,64,82)(4,38,65,75)(5,31,66,92)(6,48,67,85)(7,41,68,78)(8,34,69,95)(9,27,70,88)(10,44,71,81)(11,37,72,74)(12,30,49,91)(13,47,50,84)(14,40,51,77)(15,33,52,94)(16,26,53,87)(17,43,54,80)(18,36,55,73)(19,29,56,90)(20,46,57,83)(21,39,58,76)(22,32,59,93)(23,25,60,86)(24,42,61,79), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,94)(26,75)(27,80)(28,85)(29,90)(30,95)(31,76)(32,81)(33,86)(34,91)(35,96)(36,77)(37,82)(38,87)(39,92)(40,73)(41,78)(42,83)(43,88)(44,93)(45,74)(46,79)(47,84)(48,89)(49,69)(51,55)(52,60)(53,65)(54,70)(57,61)(58,66)(59,71)(63,67)(64,72)>;
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), (1,35,62,96)(2,28,63,89)(3,45,64,82)(4,38,65,75)(5,31,66,92)(6,48,67,85)(7,41,68,78)(8,34,69,95)(9,27,70,88)(10,44,71,81)(11,37,72,74)(12,30,49,91)(13,47,50,84)(14,40,51,77)(15,33,52,94)(16,26,53,87)(17,43,54,80)(18,36,55,73)(19,29,56,90)(20,46,57,83)(21,39,58,76)(22,32,59,93)(23,25,60,86)(24,42,61,79), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,94)(26,75)(27,80)(28,85)(29,90)(30,95)(31,76)(32,81)(33,86)(34,91)(35,96)(36,77)(37,82)(38,87)(39,92)(40,73)(41,78)(42,83)(43,88)(44,93)(45,74)(46,79)(47,84)(48,89)(49,69)(51,55)(52,60)(53,65)(54,70)(57,61)(58,66)(59,71)(63,67)(64,72) );
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)], [(1,35,62,96),(2,28,63,89),(3,45,64,82),(4,38,65,75),(5,31,66,92),(6,48,67,85),(7,41,68,78),(8,34,69,95),(9,27,70,88),(10,44,71,81),(11,37,72,74),(12,30,49,91),(13,47,50,84),(14,40,51,77),(15,33,52,94),(16,26,53,87),(17,43,54,80),(18,36,55,73),(19,29,56,90),(20,46,57,83),(21,39,58,76),(22,32,59,93),(23,25,60,86),(24,42,61,79)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,94),(26,75),(27,80),(28,85),(29,90),(30,95),(31,76),(32,81),(33,86),(34,91),(35,96),(36,77),(37,82),(38,87),(39,92),(40,73),(41,78),(42,83),(43,88),(44,93),(45,74),(46,79),(47,84),(48,89),(49,69),(51,55),(52,60),(53,65),(54,70),(57,61),(58,66),(59,71),(63,67),(64,72)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | M4(2) | C4○D4 | C3⋊D4 | C4×S3 | C4×S3 | C8○D4 | C4○D12 | S3×M4(2) | D12.C4 |
| kernel | C24⋊21D4 | C8×Dic3 | Dic3⋊C8 | D6⋊C8 | C12.55D4 | C2×C8⋊S3 | C4×C3⋊D4 | C6×M4(2) | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C2×M4(2) | C24 | C2×C8 | C22×C4 | Dic3 | C12 | C8 | C2×C4 | C23 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C24⋊21D4 ►in GL6(𝔽73)
| 0 | 46 | 0 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 3 | 0 | 0 |
| 0 | 0 | 40 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 48 |
| 0 | 0 | 0 | 0 | 42 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 69 | 63 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 70 | 72 |
G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,46,0,0,0,0,0,0,46,40,0,0,0,0,3,27,0,0,0,0,0,0,72,42,0,0,0,0,48,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,10,69,0,0,0,0,7,63,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72,0,0,0,0,0,0,1,70,0,0,0,0,0,72] >;
C24⋊21D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{21}D_4 % in TeX
G:=Group("C24:21D4"); // GroupNames label
G:=SmallGroup(192,687);
// by ID
G=gap.SmallGroup(192,687);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations