metabelian, supersoluble, monomial
Aliases: C24⋊4D6, D24⋊4S3, D12⋊1D6, C8⋊7S32, C3⋊S3⋊2D8, C3⋊2(S3×D8), C32⋊4(C2×D8), (C3×D24)⋊9C2, C6.29(S3×D4), D6⋊D6⋊1C2, (C3×C24)⋊7C22, C32⋊2D8⋊2C2, (C3×D12)⋊3C22, C3⋊Dic3.40D4, C2.6(D6⋊D6), (C3×C12).45C23, C12.68(C22×S3), C32⋊4C8⋊17C22, (C8×C3⋊S3)⋊1C2, C4.66(C2×S32), (C2×C3⋊S3).41D4, (C3×C6).29(C2×D4), (C4×C3⋊S3).66C22, SmallGroup(288,445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊4D6
G = < a,b,c | a24=b6=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >
Subgroups: 882 in 163 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, C2×D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×D8, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C32⋊4C8, C3×C24, D6⋊S3, C3×D12, C4×C3⋊S3, C2×S32, S3×D8, C32⋊2D8, C3×D24, C8×C3⋊S3, D6⋊D6, C24⋊4D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S32, S3×D4, C2×S32, S3×D8, D6⋊D6, C24⋊4D6
►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 44 17 28 9 36)(2 43 18 27 10 35)(3 42 19 26 11 34)(4 41 20 25 12 33)(5 40 21 48 13 32)(6 39 22 47 14 31)(7 38 23 46 15 30)(8 37 24 45 16 29)
(1 5)(2 22)(3 15)(4 8)(6 18)(7 11)(9 21)(10 14)(12 24)(13 17)(16 20)(19 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 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), (1,44,17,28,9,36)(2,43,18,27,10,35)(3,42,19,26,11,34)(4,41,20,25,12,33)(5,40,21,48,13,32)(6,39,22,47,14,31)(7,38,23,46,15,30)(8,37,24,45,16,29), (1,5)(2,22)(3,15)(4,8)(6,18)(7,11)(9,21)(10,14)(12,24)(13,17)(16,20)(19,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46)>;
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), (1,44,17,28,9,36)(2,43,18,27,10,35)(3,42,19,26,11,34)(4,41,20,25,12,33)(5,40,21,48,13,32)(6,39,22,47,14,31)(7,38,23,46,15,30)(8,37,24,45,16,29), (1,5)(2,22)(3,15)(4,8)(6,18)(7,11)(9,21)(10,14)(12,24)(13,17)(16,20)(19,23)(25,45)(26,38)(27,31)(28,48)(29,41)(30,34)(32,44)(33,37)(35,47)(36,40)(39,43)(42,46) );
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)], [(1,44,17,28,9,36),(2,43,18,27,10,35),(3,42,19,26,11,34),(4,41,20,25,12,33),(5,40,21,48,13,32),(6,39,22,47,14,31),(7,38,23,46,15,30),(8,37,24,45,16,29)], [(1,5),(2,22),(3,15),(4,8),(6,18),(7,11),(9,21),(10,14),(12,24),(13,17),(16,20),(19,23),(25,45),(26,38),(27,31),(28,48),(29,41),(30,34),(32,44),(33,37),(35,47),(36,40),(39,43),(42,46)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 2 | 18 | 2 | 2 | 4 | 24 | 24 | 24 | 24 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 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 | D8 | S32 | S3×D4 | C2×S32 | S3×D8 | D6⋊D6 | C24⋊4D6 |
| kernel | C24⋊4D6 | C32⋊2D8 | C3×D24 | C8×C3⋊S3 | D6⋊D6 | D24 | C3⋊Dic3 | C2×C3⋊S3 | C24 | D12 | 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⋊4D6 ►in GL6(𝔽73)
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 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 |
| 14 | 43 | 0 | 0 | 0 | 0 |
| 43 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[14,43,0,0,0,0,43,59,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C24⋊4D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes_4D_6 % in TeX
G:=Group("C24:4D6"); // GroupNames label
G:=SmallGroup(288,445);
// by ID
G=gap.SmallGroup(288,445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,142,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations