metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.54D4, M4(2).37D6, D4⋊S3⋊6C4, C8○D4⋊11S3, D4.S3⋊6C4, D4.8(C4×S3), C3⋊Q16⋊6C4, C6.82(C4×D4), C8○D12⋊14C2, C3⋊4(C8.26D4), Q8⋊2S3⋊6C4, C4○D4.51D6, C24⋊C4⋊29C2, (C2×C8).191D6, Q8.13(C4×S3), D12.19(C2×C4), C12.448(C2×D4), C8.47(C3⋊D4), D12⋊C4⋊14C2, Q8⋊3Dic3⋊3C2, C12.30(C22×C4), Dic6.19(C2×C4), Q8.13D6.3C2, C12.53D4⋊14C2, (C2×C12).425C23, (C2×C24).280C22, C4○D12.45C22, C22.4(C4○D12), (C4×Dic3).41C22, C4.Dic3.45C22, (C3×M4(2)).40C22, C3⋊C8.6(C2×C4), C4.30(S3×C2×C4), (C3×C8○D4)⋊12C2, C2.27(C4×C3⋊D4), (C3×D4).15(C2×C4), C4.139(C2×C3⋊D4), (C3×Q8).15(C2×C4), (C2×C6).10(C4○D4), (C2×C3⋊C8).144C22, (C2×C4).515(C22×S3), (C3×C4○D4).40C22, SmallGroup(192,704)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.54D4
G = < a,b,c | a24=c2=1, b4=a12, bab-1=a5, cac=a17, cbc=a12b3 >
Subgroups: 240 in 104 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, Dic3 [×2], C12 [×2], C12, D6, C2×C6, C2×C6, C42, C2×C8, C2×C8 [×3], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8⋊C4, C4≀C2 [×2], 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, C8.26D4, C24⋊C4, C12.53D4, D12⋊C4, Q8⋊3Dic3, C8○D12, Q8.13D6, C3×C8○D4, C24.54D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C8.26D4, C4×C3⋊D4, C24.54D4
►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 19 10 13 4 7 22)(2 21 20 15 14 9 8 3)(5 12 23 6 17 24 11 18)(25 46 31 28 37 34 43 40)(26 27 32 33 38 39 44 45)(29 42 35 48 41 30 47 36)
(1 25)(2 42)(3 35)(4 28)(5 45)(6 38)(7 31)(8 48)(9 41)(10 34)(11 27)(12 44)(13 37)(14 30)(15 47)(16 40)(17 33)(18 26)(19 43)(20 36)(21 29)(22 46)(23 39)(24 32)
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), (1,16,19,10,13,4,7,22)(2,21,20,15,14,9,8,3)(5,12,23,6,17,24,11,18)(25,46,31,28,37,34,43,40)(26,27,32,33,38,39,44,45)(29,42,35,48,41,30,47,36), (1,25)(2,42)(3,35)(4,28)(5,45)(6,38)(7,31)(8,48)(9,41)(10,34)(11,27)(12,44)(13,37)(14,30)(15,47)(16,40)(17,33)(18,26)(19,43)(20,36)(21,29)(22,46)(23,39)(24,32)>;
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), (1,16,19,10,13,4,7,22)(2,21,20,15,14,9,8,3)(5,12,23,6,17,24,11,18)(25,46,31,28,37,34,43,40)(26,27,32,33,38,39,44,45)(29,42,35,48,41,30,47,36), (1,25)(2,42)(3,35)(4,28)(5,45)(6,38)(7,31)(8,48)(9,41)(10,34)(11,27)(12,44)(13,37)(14,30)(15,47)(16,40)(17,33)(18,26)(19,43)(20,36)(21,29)(22,46)(23,39)(24,32) );
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)], [(1,16,19,10,13,4,7,22),(2,21,20,15,14,9,8,3),(5,12,23,6,17,24,11,18),(25,46,31,28,37,34,43,40),(26,27,32,33,38,39,44,45),(29,42,35,48,41,30,47,36)], [(1,25),(2,42),(3,35),(4,28),(5,45),(6,38),(7,31),(8,48),(9,41),(10,34),(11,27),(12,44),(13,37),(14,30),(15,47),(16,40),(17,33),(18,26),(19,43),(20,36),(21,29),(22,46),(23,39),(24,32)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 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 | 6 | 6 | 6 | 6 | 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 | 12 | 12 | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 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 | 4 | 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 | C8.26D4 | C24.54D4 |
| kernel | C24.54D4 | C24⋊C4 | 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 | 2 | 4 |
Matrix representation of C24.54D4 ►in GL4(𝔽5) generated by
| 2 | 4 | 2 | 3 |
| 4 | 4 | 3 | 2 |
| 2 | 4 | 1 | 2 |
| 0 | 2 | 1 | 3 |
| 1 | 4 | 3 | 3 |
| 1 | 2 | 2 | 0 |
| 4 | 3 | 2 | 1 |
| 1 | 1 | 3 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 |
G:=sub<GL(4,GF(5))| [2,4,2,0,4,4,4,2,2,3,1,1,3,2,2,3],[1,1,4,1,4,2,3,1,3,2,2,3,3,0,1,0],[4,0,0,0,0,0,0,4,0,0,1,0,0,4,0,0] >;
C24.54D4 in GAP, Magma, Sage, TeX
C_{24}._{54}D_4 % in TeX
G:=Group("C24.54D4"); // GroupNames label
G:=SmallGroup(192,704);
// by ID
G=gap.SmallGroup(192,704);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,136,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^4=a^12,b*a*b^-1=a^5,c*a*c=a^17,c*b*c=a^12*b^3>;
// generators/relations