metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.100D4, M4(2).36D6, D4⋊S3⋊7C4, C3⋊4(C8○D8), C8○D4⋊10S3, D4.S3⋊7C4, C3⋊Q16⋊7C4, D4.7(C4×S3), C6.81(C4×D4), C8○D12⋊13C2, Q8⋊2S3⋊7C4, C4○D4.50D6, (C2×C8).278D6, Q8.12(C4×S3), (C8×Dic3)⋊32C2, D12.18(C2×C4), C12.447(C2×D4), C8.22(C3⋊D4), D12⋊C4⋊15C2, C12.29(C22×C4), Dic6.18(C2×C4), Q8.13D6.4C2, Q8⋊3Dic3⋊15C2, C12.53D4⋊15C2, (C2×C24).279C22, (C2×C12).424C23, C4○D12.44C22, C22.3(C4○D12), C4.Dic3.44C22, (C4×Dic3).236C22, (C3×M4(2)).39C22, C4.29(S3×C2×C4), C3⋊C8.10(C2×C4), (C3×C8○D4)⋊11C2, C2.26(C4×C3⋊D4), (C3×D4).14(C2×C4), (C2×C6).9(C4○D4), C4.138(C2×C3⋊D4), (C3×Q8).14(C2×C4), (C2×C3⋊C8).267C22, (C2×C4).514(C22×S3), (C3×C4○D4).39C22, SmallGroup(192,703)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.100D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=cac=a17, cbc=a12b3 >
Subgroups: 240 in 106 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4×C8, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C4○D12, C3×C4○D4, C8○D8, C8×Dic3, C12.53D4, D12⋊C4, Q8⋊3Dic3, C8○D12, Q8.13D6, C3×C8○D4, C24.100D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C8○D8, C4×C3⋊D4, C24.100D4
►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 16 7 22 13 4 19 10)(2 9 8 15 14 21 20 3)(5 12 11 18 17 24 23 6)(25 34 43 28 37 46 31 40)(26 27 44 45 38 39 32 33)(29 30 47 48 41 42 35 36)
(1 46)(2 39)(3 32)(4 25)(5 42)(6 35)(7 28)(8 45)(9 38)(10 31)(11 48)(12 41)(13 34)(14 27)(15 44)(16 37)(17 30)(18 47)(19 40)(20 33)(21 26)(22 43)(23 36)(24 29)
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), (1,16,7,22,13,4,19,10)(2,9,8,15,14,21,20,3)(5,12,11,18,17,24,23,6)(25,34,43,28,37,46,31,40)(26,27,44,45,38,39,32,33)(29,30,47,48,41,42,35,36), (1,46)(2,39)(3,32)(4,25)(5,42)(6,35)(7,28)(8,45)(9,38)(10,31)(11,48)(12,41)(13,34)(14,27)(15,44)(16,37)(17,30)(18,47)(19,40)(20,33)(21,26)(22,43)(23,36)(24,29)>;
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), (1,16,7,22,13,4,19,10)(2,9,8,15,14,21,20,3)(5,12,11,18,17,24,23,6)(25,34,43,28,37,46,31,40)(26,27,44,45,38,39,32,33)(29,30,47,48,41,42,35,36), (1,46)(2,39)(3,32)(4,25)(5,42)(6,35)(7,28)(8,45)(9,38)(10,31)(11,48)(12,41)(13,34)(14,27)(15,44)(16,37)(17,30)(18,47)(19,40)(20,33)(21,26)(22,43)(23,36)(24,29) );
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)], [(1,16,7,22,13,4,19,10),(2,9,8,15,14,21,20,3),(5,12,11,18,17,24,23,6),(25,34,43,28,37,46,31,40),(26,27,44,45,38,39,32,33),(29,30,47,48,41,42,35,36)], [(1,46),(2,39),(3,32),(4,25),(5,42),(6,35),(7,28),(8,45),(9,38),(10,31),(11,48),(12,41),(13,34),(14,27),(15,44),(16,37),(17,30),(18,47),(19,40),(20,33),(21,26),(22,43),(23,36),(24,29)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 1 | 1 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4×S3 | C4○D12 | C8○D8 | C24.100D4 |
| kernel | C24.100D4 | C8×Dic3 | C12.53D4 | D12⋊C4 | Q8⋊3Dic3 | C8○D12 | Q8.13D6 | C3×C8○D4 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | C8○D4 | C24 | C2×C8 | M4(2) | C4○D4 | C2×C6 | C8 | D4 | Q8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C24.100D4 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 22 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,22,0,0,0,0,22],[72,1,0,0,0,1,0,0,0,0,10,0,0,0,0,22],[1,72,0,0,0,72,0,0,0,0,0,10,0,0,22,0] >;
C24.100D4 in GAP, Magma, Sage, TeX
C_{24}._{100}D_4 % in TeX
G:=Group("C24.100D4"); // GroupNames label
G:=SmallGroup(192,703);
// by ID
G=gap.SmallGroup(192,703);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,58,136,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=c*a*c=a^17,c*b*c=a^12*b^3>;
// generators/relations