metabelian, supersoluble, monomial, A-group
Aliases: C24.60D6, C3⋊C16⋊6S3, C8.21S32, C3⋊S3⋊2C16, C3⋊1(S3×C16), C6.1(S3×C8), C32⋊5(C2×C16), C12.43(C4×S3), C3⋊Dic3.3C8, C32⋊4C8.4C4, (C3×C24).42C22, C4.11(C6.D6), C2.1(C12.29D6), (C3×C3⋊C16)⋊8C2, (C8×C3⋊S3).3C2, (C4×C3⋊S3).7C4, (C2×C3⋊S3).3C8, (C3×C6).13(C2×C8), (C3×C12).78(C2×C4), SmallGroup(288,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C24.60D6 |
Generators and relations for C24.60D6
G = < a,b,c | a24=1, b6=a3, c2=a18, bab-1=cac-1=a17, cbc-1=a6b5 >
Subgroups: 210 in 67 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, S3×C8, C32⋊4C8, C3×C24, C4×C3⋊S3, S3×C16, C3×C3⋊C16, C8×C3⋊S3, C24.60D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C16, C2×C8, C4×S3, C2×C16, S32, S3×C8, C6.D6, S3×C16, C12.29D6, C24.60D6
►On 48 points
(1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47)(2 36 22 8 42 28 14 48 34 20 6 40 26 12 46 32 18 4 38 24 10 44 30 16)
(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 3 37 39 25 27 13 15)(2 20 38 8 26 44 14 32)(4 6 40 42 28 30 16 18)(5 23 41 11 29 47 17 35)(7 9 43 45 31 33 19 21)(10 12 46 48 34 36 22 24)
G:=sub<Sym(48)| (1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47)(2,36,22,8,42,28,14,48,34,20,6,40,26,12,46,32,18,4,38,24,10,44,30,16), (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), (1,3,37,39,25,27,13,15)(2,20,38,8,26,44,14,32)(4,6,40,42,28,30,16,18)(5,23,41,11,29,47,17,35)(7,9,43,45,31,33,19,21)(10,12,46,48,34,36,22,24)>;
G:=Group( (1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47)(2,36,22,8,42,28,14,48,34,20,6,40,26,12,46,32,18,4,38,24,10,44,30,16), (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), (1,3,37,39,25,27,13,15)(2,20,38,8,26,44,14,32)(4,6,40,42,28,30,16,18)(5,23,41,11,29,47,17,35)(7,9,43,45,31,33,19,21)(10,12,46,48,34,36,22,24) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47),(2,36,22,8,42,28,14,48,34,20,6,40,26,12,46,32,18,4,38,24,10,44,30,16)], [(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)], [(1,3,37,39,25,27,13,15),(2,20,38,8,26,44,14,32),(4,6,40,42,28,30,16,18),(5,23,41,11,29,47,17,35),(7,9,43,45,31,33,19,21),(10,12,46,48,34,36,22,24)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16P | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 4 | 4 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | C4×S3 | S3×C8 | S3×C16 | S32 | C6.D6 | C12.29D6 | C24.60D6 |
| kernel | C24.60D6 | C3×C3⋊C16 | C8×C3⋊S3 | C32⋊4C8 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C3⋊C16 | C24 | C12 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C24.60D6 ►in GL4(𝔽97) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 96 | 1 |
| 0 | 0 | 96 | 0 |
| 12 | 85 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 47 | 0 | 0 |
| 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(97))| [47,0,0,0,0,47,0,0,0,0,96,96,0,0,1,0],[12,12,0,0,85,0,0,0,0,0,0,1,0,0,1,0],[0,47,0,0,47,0,0,0,0,0,0,1,0,0,1,0] >;
C24.60D6 in GAP, Magma, Sage, TeX
C_{24}._{60}D_6 % in TeX
G:=Group("C24.60D6"); // GroupNames label
G:=SmallGroup(288,190);
// by ID
G=gap.SmallGroup(288,190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,36,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=a^3,c^2=a^18,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^6*b^5>;
// generators/relations