metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊11C4, C8.14D12, C24.74D4, Dic12⋊11C4, C42.267D6, (C4×C8)⋊10S3, C3⋊1(C8○D8), C6.9(C4×D4), C24⋊C2⋊7C4, (C4×C24)⋊15C2, C8.22(C4×S3), C8○D12⋊11C2, C24.52(C2×C4), (C2×C8).324D6, C4.76(C2×D12), C2.12(C4×D12), C4○D24.10C2, D12.13(C2×C4), C12.296(C2×D4), C24.C4⋊17C2, C42⋊4S3⋊15C2, Dic6.13(C2×C4), (C2×C24).426C22, C12.104(C22×C4), (C4×C12).328C22, (C2×C12).791C23, C4○D12.35C22, C22.20(C4○D12), C4.Dic3.33C22, C4.62(S3×C2×C4), (C2×C6).62(C4○D4), (C2×C4).672(C22×S3), SmallGroup(192,259)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊11C4
G = < a,b,c | a24=b2=c4=1, bab=a-1, ac=ca, cbc-1=a18b >
Subgroups: 264 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, 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, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C4×C8, 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○D8, C42⋊4S3, C24.C4, C4×C24, C8○D12, C4○D24, D24⋊11C4
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○D8, C4×D12, D24⋊11C4
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 7 13 19)(2 8 14 20)(3 9 15 21)(4 10 16 22)(5 11 17 23)(6 12 18 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)], [(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | C8○D8 | D24⋊11C4 |
| kernel | D24⋊11C4 | C42⋊4S3 | C24.C4 | C4×C24 | C8○D12 | C4○D24 | C24⋊C2 | D24 | Dic12 | C4×C8 | 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 | 8 | 16 |
Matrix representation of D24⋊11C4 ►in GL2(𝔽73) generated by
| 17 | 0 |
| 0 | 43 |
| 0 | 43 |
| 17 | 0 |
| 46 | 0 |
| 0 | 72 |
G:=sub<GL(2,GF(73))| [17,0,0,43],[0,17,43,0],[46,0,0,72] >;
D24⋊11C4 in GAP, Magma, Sage, TeX
D_{24}\rtimes_{11}C_4 % in TeX
G:=Group("D24:11C4"); // GroupNames label
G:=SmallGroup(192,259);
// by ID
G=gap.SmallGroup(192,259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,58,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^18*b>;
// generators/relations