metabelian, supersoluble, monomial
Aliases: C24.23D6, Dic12⋊4S3, Dic6.1D6, C8.16S32, C3⋊S3⋊2Q16, C3⋊2(S3×Q16), C6.31(S3×D4), C32⋊4(C2×Q16), (C3×Dic12)⋊9C2, C3⋊Dic3.41D4, C32⋊2Q16⋊2C2, C2.8(D6⋊D6), (C3×C12).50C23, (C3×C24).22C22, C12.70(C22×S3), Dic3.D6.3C2, (C3×Dic6).4C22, C32⋊4C8.21C22, C4.68(C2×S32), (C8×C3⋊S3).1C2, (C2×C3⋊S3).42D4, (C3×C6).34(C2×D4), (C4×C3⋊S3).67C22, SmallGroup(288,450)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.23D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=a-1, cac=a17, cbc=b5 >
Subgroups: 498 in 131 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, Q8, C32, Dic3, C12, C12, D6, C2×C8, Q16, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C3×Q8, C2×Q16, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, C32⋊4C8, C3×C24, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, S3×Q16, C32⋊2Q16, C3×Dic12, C8×C3⋊S3, Dic3.D6, C24.23D6
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C22×S3, C2×Q16, S32, S3×D4, C2×S32, S3×Q16, D6⋊D6, C24.23D6
►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 32 5 28 9 48 13 44 17 40 21 36)(2 31 6 27 10 47 14 43 18 39 22 35)(3 30 7 26 11 46 15 42 19 38 23 34)(4 29 8 25 12 45 16 41 20 37 24 33)
(1 17)(2 10)(4 20)(5 13)(7 23)(8 16)(11 19)(14 22)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(40 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), (1,32,5,28,9,48,13,44,17,40,21,36)(2,31,6,27,10,47,14,43,18,39,22,35)(3,30,7,26,11,46,15,42,19,38,23,34)(4,29,8,25,12,45,16,41,20,37,24,33), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48)>;
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), (1,32,5,28,9,48,13,44,17,40,21,36)(2,31,6,27,10,47,14,43,18,39,22,35)(3,30,7,26,11,46,15,42,19,38,23,34)(4,29,8,25,12,45,16,41,20,37,24,33), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48) );
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)], [(1,32,5,28,9,48,13,44,17,40,21,36),(2,31,6,27,10,47,14,43,18,39,22,35),(3,30,7,26,11,46,15,42,19,38,23,34),(4,29,8,25,12,45,16,41,20,37,24,33)], [(1,17),(2,10),(4,20),(5,13),(7,23),(8,16),(11,19),(14,22),(25,33),(27,43),(28,36),(30,46),(31,39),(34,42),(37,45),(40,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 2 | 12 | 12 | 12 | 12 | 18 | 2 | 2 | 4 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | S32 | S3×D4 | C2×S32 | S3×Q16 | D6⋊D6 | C24.23D6 |
| kernel | C24.23D6 | C32⋊2Q16 | C3×Dic12 | C8×C3⋊S3 | Dic3.D6 | Dic12 | C3⋊Dic3 | C2×C3⋊S3 | C24 | Dic6 | C3⋊S3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 4 |
Matrix representation of C24.23D6 ►in GL6(𝔽73)
| 0 | 41 | 0 | 0 | 0 | 0 |
| 16 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 2 | 31 | 0 | 0 | 0 | 0 |
| 54 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[2,54,0,0,0,0,31,71,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C24.23D6 in GAP, Magma, Sage, TeX
C_{24}._{23}D_6 % in TeX
G:=Group("C24.23D6"); // GroupNames label
G:=SmallGroup(288,450);
// by ID
G=gap.SmallGroup(288,450);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,142,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^5>;
// generators/relations