metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24:20D4, D6:4M4(2), D6:C8:38C2, C3:7(C8:9D4), C6.79(C4xD4), C8:12(C3:D4), D6:C4.16C4, C24:C4:28C2, (C2xC8).188D6, Dic3:C8:40C2, C6.32(C8oD4), C12.442(C2xD4), (C2xM4(2)):7S3, C23.24(C4xS3), (C6xM4(2)):11C2, Dic3:C4.16C4, (C22xC4).152D6, C2.21(S3xM4(2)), C6.32(C2xM4(2)), C12.253(C4oD4), C4.137(C4oD12), C12.55D4:29C2, (C2xC24).277C22, (C2xC12).867C23, C2.17(D12.C4), C6.D4.12C4, (C22xC12).374C22, (C4xDic3).189C22, (S3xC2xC8):27C2, (C2xC4).50(C4xS3), C2.24(C4xC3:D4), (C4xC3:D4).16C2, (C2xC3:D4).10C4, C4.133(C2xC3:D4), C22.146(S3xC2xC4), (C2xC12).187(C2xC4), (C2xC3:C8).323C22, (S3xC2xC4).285C22, (C22xC6).67(C2xC4), (C22xS3).44(C2xC4), (C2xC4).809(C22xS3), (C2xC6).137(C22xC4), (C2xDic3).34(C2xC4), SmallGroup(192,686)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:D4
G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=a17, cbc=b-1 >
Subgroups: 280 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C3:C8, C24, C24, C4xS3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xC6, C8:C4, C22:C8, C4:C8, C4xD4, C22xC8, C2xM4(2), S3xC8, C2xC3:C8, C4xDic3, Dic3:C4, D6:C4, C6.D4, C2xC24, C3xM4(2), S3xC2xC4, C2xC3:D4, C22xC12, C8:9D4, Dic3:C8, C24:C4, D6:C8, C12.55D4, S3xC2xC8, C4xC3:D4, C6xM4(2), C24:D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, M4(2), C22xC4, C2xD4, C4oD4, C4xS3, C3:D4, C22xS3, C4xD4, C2xM4(2), C8oD4, S3xC2xC4, C4oD12, C2xC3:D4, C8:9D4, S3xM4(2), D12.C4, C4xC3:D4, C24:D4
►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 70 38 92)(2 51 39 73)(3 56 40 78)(4 61 41 83)(5 66 42 88)(6 71 43 93)(7 52 44 74)(8 57 45 79)(9 62 46 84)(10 67 47 89)(11 72 48 94)(12 53 25 75)(13 58 26 80)(14 63 27 85)(15 68 28 90)(16 49 29 95)(17 54 30 76)(18 59 31 81)(19 64 32 86)(20 69 33 91)(21 50 34 96)(22 55 35 77)(23 60 36 82)(24 65 37 87)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(40 48)(49 95)(50 88)(51 81)(52 74)(53 91)(54 84)(55 77)(56 94)(57 87)(58 80)(59 73)(60 90)(61 83)(62 76)(63 93)(64 86)(65 79)(66 96)(67 89)(68 82)(69 75)(70 92)(71 85)(72 78)
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), (1,70,38,92)(2,51,39,73)(3,56,40,78)(4,61,41,83)(5,66,42,88)(6,71,43,93)(7,52,44,74)(8,57,45,79)(9,62,46,84)(10,67,47,89)(11,72,48,94)(12,53,25,75)(13,58,26,80)(14,63,27,85)(15,68,28,90)(16,49,29,95)(17,54,30,76)(18,59,31,81)(19,64,32,86)(20,69,33,91)(21,50,34,96)(22,55,35,77)(23,60,36,82)(24,65,37,87), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48)(49,95)(50,88)(51,81)(52,74)(53,91)(54,84)(55,77)(56,94)(57,87)(58,80)(59,73)(60,90)(61,83)(62,76)(63,93)(64,86)(65,79)(66,96)(67,89)(68,82)(69,75)(70,92)(71,85)(72,78)>;
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), (1,70,38,92)(2,51,39,73)(3,56,40,78)(4,61,41,83)(5,66,42,88)(6,71,43,93)(7,52,44,74)(8,57,45,79)(9,62,46,84)(10,67,47,89)(11,72,48,94)(12,53,25,75)(13,58,26,80)(14,63,27,85)(15,68,28,90)(16,49,29,95)(17,54,30,76)(18,59,31,81)(19,64,32,86)(20,69,33,91)(21,50,34,96)(22,55,35,77)(23,60,36,82)(24,65,37,87), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48)(49,95)(50,88)(51,81)(52,74)(53,91)(54,84)(55,77)(56,94)(57,87)(58,80)(59,73)(60,90)(61,83)(62,76)(63,93)(64,86)(65,79)(66,96)(67,89)(68,82)(69,75)(70,92)(71,85)(72,78) );
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)], [(1,70,38,92),(2,51,39,73),(3,56,40,78),(4,61,41,83),(5,66,42,88),(6,71,43,93),(7,52,44,74),(8,57,45,79),(9,62,46,84),(10,67,47,89),(11,72,48,94),(12,53,25,75),(13,58,26,80),(14,63,27,85),(15,68,28,90),(16,49,29,95),(17,54,30,76),(18,59,31,81),(19,64,32,86),(20,69,33,91),(21,50,34,96),(22,55,35,77),(23,60,36,82),(24,65,37,87)], [(2,18),(3,11),(5,21),(6,14),(8,24),(9,17),(12,20),(15,23),(25,33),(27,43),(28,36),(30,46),(31,39),(34,42),(37,45),(40,48),(49,95),(50,88),(51,81),(52,74),(53,91),(54,84),(55,77),(56,94),(57,87),(58,80),(59,73),(60,90),(61,83),(62,76),(63,93),(64,86),(65,79),(66,96),(67,89),(68,82),(69,75),(70,92),(71,85),(72,78)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 2 | 3 | 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 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 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 | C4oD4 | M4(2) | C3:D4 | C4xS3 | C4xS3 | C8oD4 | C4oD12 | S3xM4(2) | D12.C4 |
| kernel | C24:D4 | Dic3:C8 | C24:C4 | D6:C8 | C12.55D4 | S3xC2xC8 | C4xC3:D4 | C6xM4(2) | Dic3:C4 | D6:C4 | C6.D4 | C2xC3:D4 | C2xM4(2) | C24 | C2xC8 | C22xC4 | C12 | D6 | C8 | C2xC4 | C23 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C24:D4 ►in GL6(F73)
| 27 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 71 | 0 | 0 |
| 0 | 0 | 13 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 12 |
| 0 | 0 | 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 21 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 47 |
| 0 | 0 | 0 | 0 | 48 | 58 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 45 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [27,46,0,0,0,0,27,0,0,0,0,0,0,0,27,13,0,0,0,0,71,46,0,0,0,0,0,0,10,0,0,0,0,0,12,63],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,21,1,0,0,0,0,0,0,15,48,0,0,0,0,47,58],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,45,72] >;
C24:D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes D_4 % in TeX
G:=Group("C24:D4"); // GroupNames label
G:=SmallGroup(192,686);
// by ID
G=gap.SmallGroup(192,686);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations