metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊10C4, Dic12⋊10C4, M4(2).26D6, C3⋊C8.35D4, C24⋊C2⋊4C4, C8.11(C4×S3), C24⋊C4⋊1C2, (C2×C8).69D6, C6.55(C4×D4), C3⋊3(C8.26D4), C8.C4⋊4S3, C24.29(C2×C4), C4○D24.4C2, C4.212(S3×D4), D12.C4⋊10C2, D12.10(C2×C4), C12.371(C2×D4), D12⋊C4⋊11C2, C12.54(C22×C4), (C2×C24).41C22, Dic6.10(C2×C4), (C2×C12).310C23, C4○D12.17C22, C2.15(Dic3⋊5D4), C22.1(Q8⋊3S3), (C4×Dic3).40C22, (C3×M4(2)).28C22, C4.46(S3×C2×C4), (C3×C8.C4)⋊4C2, (C2×C6).1(C4○D4), (C2×C3⋊C8).78C22, (C2×C4).413(C22×S3), SmallGroup(192,453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊10C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a10b >
Subgroups: 272 in 104 conjugacy classes, 45 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C2×C3⋊C8, C4×Dic3, C2×C24, C3×M4(2), C4○D12, C8.26D4, C24⋊C4, D12⋊C4, C3×C8.C4, C4○D24, D12.C4, D24⋊10C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S3×C2×C4, S3×D4, Q8⋊3S3, C8.26D4, Dic3⋊5D4, D24⋊10C4
►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 28)(2 27)(3 26)(4 25)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)
(1 19 13 7)(2 24 14 12)(3 5 15 17)(4 10 16 22)(6 20 18 8)(9 11 21 23)(25 45)(27 31)(28 36)(29 41)(30 46)(33 37)(34 42)(35 47)(39 43)(40 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), (1,28)(2,27)(3,26)(4,25)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,19,13,7)(2,24,14,12)(3,5,15,17)(4,10,16,22)(6,20,18,8)(9,11,21,23)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48)>;
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), (1,28)(2,27)(3,26)(4,25)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29), (1,19,13,7)(2,24,14,12)(3,5,15,17)(4,10,16,22)(6,20,18,8)(9,11,21,23)(25,45)(27,31)(28,36)(29,41)(30,46)(33,37)(34,42)(35,47)(39,43)(40,48) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29)], [(1,19,13,7),(2,24,14,12),(3,5,15,17),(4,10,16,22),(6,20,18,8),(9,11,21,23),(25,45),(27,31),(28,36),(29,41),(30,46),(33,37),(34,42),(35,47),(39,43),(40,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 8A | ··· | 8F | 8G | 8H | 8I | 8J | 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 | 6 | 6 | 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 | 12 | 12 | 12 | 12 | 2 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | Q8⋊3S3 | C8.26D4 | D24⋊10C4 |
| kernel | D24⋊10C4 | C24⋊C4 | D12⋊C4 | C3×C8.C4 | C4○D24 | D12.C4 | C24⋊C2 | D24 | Dic12 | C8.C4 | C3⋊C8 | C2×C8 | M4(2) | C2×C6 | C8 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 4 |
Matrix representation of D24⋊10C4 ►in GL4(𝔽5) generated by
| 0 | 0 | 1 | 1 |
| 0 | 0 | 2 | 4 |
| 0 | 3 | 0 | 1 |
| 4 | 0 | 2 | 4 |
| 1 | 1 | 2 | 2 |
| 2 | 4 | 4 | 0 |
| 0 | 2 | 0 | 4 |
| 4 | 3 | 2 | 0 |
| 1 | 1 | 3 | 3 |
| 2 | 0 | 1 | 2 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 4 |
G:=sub<GL(4,GF(5))| [0,0,0,4,0,0,3,0,1,2,0,2,1,4,1,4],[1,2,0,4,1,4,2,3,2,4,0,2,2,0,4,0],[1,2,0,0,1,0,0,0,3,1,0,2,3,2,1,4] >;
D24⋊10C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes_{10}C_4 % in TeX
G:=Group("D24:10C4"); // GroupNames label
G:=SmallGroup(192,453);
// by ID
G=gap.SmallGroup(192,453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,219,58,136,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b>;
// generators/relations