metabelian, supersoluble, monomial
Aliases: C24.49D6, C6.14D24, C32⋊4SD32, Dic12⋊1S3, C12.11D12, C8.6S32, C3⋊C16⋊3S3, (C3×C6).10D8, C6.3(D4⋊S3), C3⋊3(C48⋊C2), (C3×C12).25D4, C32⋊5D8.2C2, (C3×Dic12)⋊6C2, C3⋊1(C8.6D6), C4.3(C3⋊D12), C2.6(C3⋊D24), C12.68(C3⋊D4), (C3×C24).10C22, (C3×C3⋊C16)⋊3C2, SmallGroup(288,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.49D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=cac=a-1, cbc=a15b5 >
Subgroups: 434 in 63 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C16, D8, Q16, C3⋊S3, C3×C6, C24, C24, Dic6, D12, C3×Q8, SD32, C3×Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, D24, Dic12, C3×Q16, C3×C24, C3×Dic6, C12⋊S3, C48⋊C2, C8.6D6, C3×C3⋊C16, C3×Dic12, C32⋊5D8, C24.49D6
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3⋊D4, SD32, S32, D24, D4⋊S3, C3⋊D12, C48⋊C2, C8.6D6, C3⋊D24, C24.49D6
►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 45 21 25 17 29 13 33 9 37 5 41)(2 44 22 48 18 28 14 32 10 36 6 40)(3 43 23 47 19 27 15 31 11 35 7 39)(4 42 24 46 20 26 16 30 12 34 8 38)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 24)(20 23)(21 22)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(42 48)(43 47)(44 46)
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), (1,45,21,25,17,29,13,33,9,37,5,41)(2,44,22,48,18,28,14,32,10,36,6,40)(3,43,23,47,19,27,15,31,11,35,7,39)(4,42,24,46,20,26,16,30,12,34,8,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(42,48)(43,47)(44,46)>;
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), (1,45,21,25,17,29,13,33,9,37,5,41)(2,44,22,48,18,28,14,32,10,36,6,40)(3,43,23,47,19,27,15,31,11,35,7,39)(4,42,24,46,20,26,16,30,12,34,8,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(42,48)(43,47)(44,46) );
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)], [(1,45,21,25,17,29,13,33,9,37,5,41),(2,44,22,48,18,28,14,32,10,36,6,40),(3,43,23,47,19,27,15,31,11,35,7,39),(4,42,24,46,20,26,16,30,12,34,8,38)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,24),(20,23),(21,22),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(42,48),(43,47),(44,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 72 | 2 | 2 | 4 | 2 | 24 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D8 | D12 | C3⋊D4 | SD32 | D24 | C48⋊C2 | S32 | D4⋊S3 | C3⋊D12 | C8.6D6 | C3⋊D24 | C24.49D6 |
| kernel | C24.49D6 | C3×C3⋊C16 | C3×Dic12 | C32⋊5D8 | C3⋊C16 | Dic12 | C3×C12 | C24 | C3×C6 | C12 | C12 | C32 | C6 | C3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C24.49D6 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 81 | 95 |
| 0 | 0 | 2 | 79 |
| 0 | 96 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 2 | 43 |
| 0 | 0 | 45 | 95 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 95 |
| 0 | 0 | 16 | 79 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,81,2,0,0,95,79],[0,1,0,0,96,1,0,0,0,0,2,45,0,0,43,95],[0,1,0,0,1,0,0,0,0,0,18,16,0,0,95,79] >;
C24.49D6 in GAP, Magma, Sage, TeX
C_{24}._{49}D_6 % in TeX
G:=Group("C24.49D6"); // GroupNames label
G:=SmallGroup(288,197);
// by ID
G=gap.SmallGroup(288,197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,92,590,58,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=c*a*c=a^-1,c*b*c=a^15*b^5>;
// generators/relations