metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24:29D4, C22:2D24, C23.30D12, (C2xC6):5D8, (C2xD24):9C2, C3:4(C8:7D4), C6.17(C2xD8), C12:7D4:2C2, C8:13(C3:D4), (C22xC8):10S3, C24:1C4:14C2, (C2xC8).308D6, C2.D24:3C2, (C2xC4).68D12, C2.17(C2xD24), (C22xC24):10C2, C6.18(C4oD8), C12.413(C2xD4), (C2xC12).356D4, C2.18(C4oD24), C6.71(C4:D4), (C22xC4).447D6, (C22xC6).141D4, C4.112(C4oD12), C12.228(C4oD4), C2.19(C12:7D4), (C2xC24).380C22, (C2xC12).769C23, (C2xD12).19C22, C22.132(C2xD12), C4:Dic3.24C22, (C22xC12).519C22, (C2xC6).159(C2xD4), C4.106(C2xC3:D4), (C2xC4).717(C22xS3), SmallGroup(192,674)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:29D4
G = < a,b,c | a24=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 488 in 134 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C2xD4, C24, C24, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xC6, D4:C4, C2.D8, C4:D4, C22xC8, C2xD8, D24, C4:Dic3, D6:C4, C2xC24, C2xC24, C2xD12, C2xC3:D4, C22xC12, C8:7D4, C24:1C4, C2.D24, C2xD24, C12:7D4, C22xC24, C24:29D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, C2xD8, C4oD8, D24, C2xD12, C4oD12, C2xC3:D4, C8:7D4, C2xD24, C4oD24, C12:7D4, C24:29D4
►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 67 36 81)(2 66 37 80)(3 65 38 79)(4 64 39 78)(5 63 40 77)(6 62 41 76)(7 61 42 75)(8 60 43 74)(9 59 44 73)(10 58 45 96)(11 57 46 95)(12 56 47 94)(13 55 48 93)(14 54 25 92)(15 53 26 91)(16 52 27 90)(17 51 28 89)(18 50 29 88)(19 49 30 87)(20 72 31 86)(21 71 32 85)(22 70 33 84)(23 69 34 83)(24 68 35 82)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(49 75)(50 74)(51 73)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)
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), (1,67,36,81)(2,66,37,80)(3,65,38,79)(4,64,39,78)(5,63,40,77)(6,62,41,76)(7,61,42,75)(8,60,43,74)(9,59,44,73)(10,58,45,96)(11,57,46,95)(12,56,47,94)(13,55,48,93)(14,54,25,92)(15,53,26,91)(16,52,27,90)(17,51,28,89)(18,50,29,88)(19,49,30,87)(20,72,31,86)(21,71,32,85)(22,70,33,84)(23,69,34,83)(24,68,35,82), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(49,75)(50,74)(51,73)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)>;
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), (1,67,36,81)(2,66,37,80)(3,65,38,79)(4,64,39,78)(5,63,40,77)(6,62,41,76)(7,61,42,75)(8,60,43,74)(9,59,44,73)(10,58,45,96)(11,57,46,95)(12,56,47,94)(13,55,48,93)(14,54,25,92)(15,53,26,91)(16,52,27,90)(17,51,28,89)(18,50,29,88)(19,49,30,87)(20,72,31,86)(21,71,32,85)(22,70,33,84)(23,69,34,83)(24,68,35,82), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(49,75)(50,74)(51,73)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76) );
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)], [(1,67,36,81),(2,66,37,80),(3,65,38,79),(4,64,39,78),(5,63,40,77),(6,62,41,76),(7,61,42,75),(8,60,43,74),(9,59,44,73),(10,58,45,96),(11,57,46,95),(12,56,47,94),(13,55,48,93),(14,54,25,92),(15,53,26,91),(16,52,27,90),(17,51,28,89),(18,50,29,88),(19,49,30,87),(20,72,31,86),(21,71,32,85),(22,70,33,84),(23,69,34,83),(24,68,35,82)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(49,75),(50,74),(51,73),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4oD4 | D8 | C3:D4 | D12 | D12 | C4oD8 | C4oD12 | D24 | C4oD24 |
| kernel | C24:29D4 | C24:1C4 | C2.D24 | C2xD24 | C12:7D4 | C22xC24 | C22xC8 | C24 | C2xC12 | C22xC6 | C2xC8 | C22xC4 | C12 | C2xC6 | C8 | C2xC4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C24:29D4 ►in GL4(F73) generated by
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [57,57,0,0,16,57,0,0,0,0,0,1,0,0,72,72],[0,27,0,0,27,0,0,0,0,0,13,43,0,0,30,60],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
C24:29D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{29}D_4 % in TeX
G:=Group("C24:29D4"); // GroupNames label
G:=SmallGroup(192,674);
// by ID
G=gap.SmallGroup(192,674);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations