metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.82D4, C22:2Dic12, C23.31D12, (C2xC6):5Q16, C24:1C4:15C2, (C2xC4).69D12, (C2xC8).309D6, C6.11(C2xQ16), (C2xDic12):9C2, C6.19(C4oD8), (C2xC12).357D4, C12.414(C2xD4), C8.39(C3:D4), C3:4(C8.18D4), (C22xC8).11S3, C2.Dic12:3C2, C2.19(C4oD24), C6.72(C4:D4), (C22xC24).15C2, C2.11(C2xDic12), (C22xC4).448D6, (C22xC6).142D4, C4.113(C4oD12), C12.229(C4oD4), C2.20(C12:7D4), (C2xC12).770C23, (C2xC24).381C22, C12.48D4.5C2, C22.133(C2xD12), C4:Dic3.25C22, (C2xDic6).18C22, (C22xC12).520C22, (C2xC6).160(C2xD4), C4.107(C2xC3:D4), (C2xC4).718(C22xS3), SmallGroup(192,675)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.82D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=cac-1=a-1, cbc-1=a12b-1 >
Subgroups: 296 in 114 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, Q16, C22xC4, C2xQ8, C24, C24, Dic6, C2xDic3, C2xC12, C2xC12, C22xC6, Q8:C4, C2.D8, C22:Q8, C22xC8, C2xQ16, Dic12, Dic3:C4, C4:Dic3, C6.D4, C2xC24, C2xC24, C2xDic6, C22xC12, C8.18D4, C2.Dic12, C24:1C4, C2xDic12, C12.48D4, C22xC24, C24.82D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, C2xQ16, C4oD8, Dic12, C2xD12, C4oD12, C2xC3:D4, C8.18D4, C4oD24, C2xDic12, C12:7D4, C24.82D4
►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 57 31 86)(2 56 32 85)(3 55 33 84)(4 54 34 83)(5 53 35 82)(6 52 36 81)(7 51 37 80)(8 50 38 79)(9 49 39 78)(10 72 40 77)(11 71 41 76)(12 70 42 75)(13 69 43 74)(14 68 44 73)(15 67 45 96)(16 66 46 95)(17 65 47 94)(18 64 48 93)(19 63 25 92)(20 62 26 91)(21 61 27 90)(22 60 28 89)(23 59 29 88)(24 58 30 87)
(1 74 13 86)(2 73 14 85)(3 96 15 84)(4 95 16 83)(5 94 17 82)(6 93 18 81)(7 92 19 80)(8 91 20 79)(9 90 21 78)(10 89 22 77)(11 88 23 76)(12 87 24 75)(25 51 37 63)(26 50 38 62)(27 49 39 61)(28 72 40 60)(29 71 41 59)(30 70 42 58)(31 69 43 57)(32 68 44 56)(33 67 45 55)(34 66 46 54)(35 65 47 53)(36 64 48 52)
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), (1,57,31,86)(2,56,32,85)(3,55,33,84)(4,54,34,83)(5,53,35,82)(6,52,36,81)(7,51,37,80)(8,50,38,79)(9,49,39,78)(10,72,40,77)(11,71,41,76)(12,70,42,75)(13,69,43,74)(14,68,44,73)(15,67,45,96)(16,66,46,95)(17,65,47,94)(18,64,48,93)(19,63,25,92)(20,62,26,91)(21,61,27,90)(22,60,28,89)(23,59,29,88)(24,58,30,87), (1,74,13,86)(2,73,14,85)(3,96,15,84)(4,95,16,83)(5,94,17,82)(6,93,18,81)(7,92,19,80)(8,91,20,79)(9,90,21,78)(10,89,22,77)(11,88,23,76)(12,87,24,75)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52)>;
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), (1,57,31,86)(2,56,32,85)(3,55,33,84)(4,54,34,83)(5,53,35,82)(6,52,36,81)(7,51,37,80)(8,50,38,79)(9,49,39,78)(10,72,40,77)(11,71,41,76)(12,70,42,75)(13,69,43,74)(14,68,44,73)(15,67,45,96)(16,66,46,95)(17,65,47,94)(18,64,48,93)(19,63,25,92)(20,62,26,91)(21,61,27,90)(22,60,28,89)(23,59,29,88)(24,58,30,87), (1,74,13,86)(2,73,14,85)(3,96,15,84)(4,95,16,83)(5,94,17,82)(6,93,18,81)(7,92,19,80)(8,91,20,79)(9,90,21,78)(10,89,22,77)(11,88,23,76)(12,87,24,75)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52) );
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)], [(1,57,31,86),(2,56,32,85),(3,55,33,84),(4,54,34,83),(5,53,35,82),(6,52,36,81),(7,51,37,80),(8,50,38,79),(9,49,39,78),(10,72,40,77),(11,71,41,76),(12,70,42,75),(13,69,43,74),(14,68,44,73),(15,67,45,96),(16,66,46,95),(17,65,47,94),(18,64,48,93),(19,63,25,92),(20,62,26,91),(21,61,27,90),(22,60,28,89),(23,59,29,88),(24,58,30,87)], [(1,74,13,86),(2,73,14,85),(3,96,15,84),(4,95,16,83),(5,94,17,82),(6,93,18,81),(7,92,19,80),(8,91,20,79),(9,90,21,78),(10,89,22,77),(11,88,23,76),(12,87,24,75),(25,51,37,63),(26,50,38,62),(27,49,39,61),(28,72,40,60),(29,71,41,59),(30,70,42,58),(31,69,43,57),(32,68,44,56),(33,67,45,55),(34,66,46,54),(35,65,47,53),(36,64,48,52)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 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 | Q16 | C3:D4 | D12 | D12 | C4oD8 | C4oD12 | Dic12 | C4oD24 |
| kernel | C24.82D4 | C2.Dic12 | C24:1C4 | C2xDic12 | C12.48D4 | C22xC24 | C22xC8 | C24 | C2xC12 | C22xC6 | C2xC8 | C22xC4 | C12 | C2xC6 | C8 | C2xC4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C24.82D4 ►in GL4(F73) generated by
| 21 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [21,0,0,0,0,7,0,0,0,0,70,0,0,0,0,24],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,1,0],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C24.82D4 in GAP, Magma, Sage, TeX
C_{24}._{82}D_4 % in TeX
G:=Group("C24.82D4"); // GroupNames label
G:=SmallGroup(192,675);
// by ID
G=gap.SmallGroup(192,675);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,254,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations