metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊4C4, C8.27D12, C24.53D4, Dic12⋊4C4, C42.23D6, C8.8(C4×S3), C24⋊C2⋊2C4, C8⋊C4⋊5S3, C24.5(C2×C4), C6.14(C4×D4), C8○D12⋊12C2, C3⋊1(C8.26D4), C4○D24.8C2, C4.78(C2×D12), C2.17(C4×D12), (C2×C8).161D6, C42⋊4S3⋊1C2, D12.15(C2×C4), C12.298(C2×D4), C24.C4⋊12C2, (C4×C12).17C22, Dic6.15(C2×C4), C12.108(C22×C4), (C2×C12).792C23, (C2×C24).272C22, C4○D12.36C22, C22.21(C4○D12), C4.Dic3.34C22, C4.66(S3×C2×C4), (C3×C8⋊C4)⋊1C2, (C2×C6).63(C4○D4), (C2×C4).682(C22×S3), SmallGroup(192,276)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊4C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a6b >
Subgroups: 264 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, C4×C12, C2×C24, C4○D12, C8.26D4, C42⋊4S3, C24.C4, C3×C8⋊C4, C8○D12, C4○D24, D24⋊4C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, S3×C2×C4, C2×D12, C4○D12, C8.26D4, C4×D12, D24⋊4C4
►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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)
(1 7 13 19)(2 20 14 8)(3 9 15 21)(4 22 16 10)(5 11 17 23)(6 24 18 12)(26 38)(28 40)(30 42)(32 44)(34 46)(36 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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34), (1,7,13,19)(2,20,14,8)(3,9,15,21)(4,22,16,10)(5,11,17,23)(6,24,18,12)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48)>;
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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34), (1,7,13,19)(2,20,14,8)(3,9,15,21)(4,22,16,10)(5,11,17,23)(6,24,18,12)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48) );
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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34)], [(1,7,13,19),(2,20,14,8),(3,9,15,21),(4,22,16,10),(5,11,17,23),(6,24,18,12),(26,38),(28,40),(30,42),(32,44),(34,46),(36,48)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | C8.26D4 | D24⋊4C4 |
| kernel | D24⋊4C4 | C42⋊4S3 | C24.C4 | C3×C8⋊C4 | C8○D12 | C4○D24 | C24⋊C2 | D24 | Dic12 | C8⋊C4 | C24 | C42 | C2×C8 | C2×C6 | C8 | C8 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of D24⋊4C4 ►in GL4(𝔽73) generated by
| 70 | 42 | 0 | 0 |
| 67 | 3 | 0 | 0 |
| 0 | 0 | 64 | 53 |
| 0 | 0 | 55 | 9 |
| 0 | 0 | 64 | 53 |
| 0 | 0 | 55 | 9 |
| 70 | 42 | 0 | 0 |
| 67 | 3 | 0 | 0 |
| 46 | 27 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [70,67,0,0,42,3,0,0,0,0,64,55,0,0,53,9],[0,0,70,67,0,0,42,3,64,55,0,0,53,9,0,0],[46,0,0,0,27,27,0,0,0,0,72,0,0,0,1,1] >;
D24⋊4C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes_4C_4 % in TeX
G:=Group("D24:4C4"); // GroupNames label
G:=SmallGroup(192,276);
// by ID
G=gap.SmallGroup(192,276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^6*b>;
// generators/relations