metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24:7C4, Dic12:7C4, M4(2).27D6, C3:3(C8oD8), C3:C8.38D4, C24:C2:6C4, C8.16(C4xS3), C6.56(C4xD4), C8.C4:8S3, C24.35(C2xC4), (C8xDic3):1C2, C4oD24.5C2, C4.213(S3xD4), (C2xC8).252D6, D12.C4:11C2, D12.11(C2xC4), C12.372(C2xD4), D12:C4:12C2, C12.55(C22xC4), (C2xC24).42C22, Dic6.11(C2xC4), (C2xC12).311C23, C4oD12.18C22, C2.16(Dic3:5D4), C22.2(Q8:3S3), (C4xDic3).235C22, (C3xM4(2)).29C22, C4.47(S3xC2xC4), (C3xC8.C4):5C2, (C2xC6).2(C4oD4), (C2xC3:C8).238C22, (C2xC4).414(C22xS3), SmallGroup(192,454)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24:7C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a17, cbc-1=a22b >
Subgroups: 272 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, D6, C2xC6, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, C3:C8, C24, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, S3xC8, C8:S3, C24:C2, D24, Dic12, C2xC3:C8, C4xDic3, C2xC24, C3xM4(2), C4oD12, C8oD8, C8xDic3, D12:C4, C3xC8.C4, C4oD24, D12.C4, D24:7C4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, C4xS3, C22xS3, C4xD4, S3xC2xC4, S3xD4, Q8:3S3, C8oD8, Dic3:5D4, D24:7C4
►On 48 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)
(1 27)(2 26)(3 25)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 19 13 7)(2 12 14 24)(3 5 15 17)(4 22 16 10)(6 8 18 20)(9 11 21 23)(25 37)(26 30)(27 47)(28 40)(29 33)(31 43)(32 36)(34 46)(35 39)(38 42)(41 45)(44 48)
G:=sub<Sym(48)| (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), (1,27)(2,26)(3,25)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,19,13,7)(2,12,14,24)(3,5,15,17)(4,22,16,10)(6,8,18,20)(9,11,21,23)(25,37)(26,30)(27,47)(28,40)(29,33)(31,43)(32,36)(34,46)(35,39)(38,42)(41,45)(44,48)>;
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), (1,27)(2,26)(3,25)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,19,13,7)(2,12,14,24)(3,5,15,17)(4,22,16,10)(6,8,18,20)(9,11,21,23)(25,37)(26,30)(27,47)(28,40)(29,33)(31,43)(32,36)(34,46)(35,39)(38,42)(41,45)(44,48) );
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)], [(1,27),(2,26),(3,25),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28)], [(1,19,13,7),(2,12,14,24),(3,5,15,17),(4,22,16,10),(6,8,18,20),(9,11,21,23),(25,37),(26,30),(27,47),(28,40),(29,33),(31,43),(32,36),(34,46),(35,39),(38,42),(41,45),(44,48)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 12A | 12B | 12C | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 4 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4oD4 | C4xS3 | C8oD8 | S3xD4 | Q8:3S3 | D24:7C4 |
| kernel | D24:7C4 | C8xDic3 | D12:C4 | C3xC8.C4 | C4oD24 | D12.C4 | C24:C2 | D24 | Dic12 | C8.C4 | C3:C8 | C2xC8 | M4(2) | C2xC6 | C8 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 4 |
Matrix representation of D24:7C4 ►in GL4(F73) generated by
| 10 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [10,0,0,0,0,22,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,72,0,0,0,0,0,0,1,0,0,1,0],[27,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
D24:7C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes_7C_4 % in TeX
G:=Group("D24:7C4"); // GroupNames label
G:=SmallGroup(192,454);
// by ID
G=gap.SmallGroup(192,454);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,555,58,136,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^22*b>;
// generators/relations