metabelian, supersoluble, monomial
Aliases: D12⋊24D6, Dic6⋊23D6, C32⋊12+ 1+4, C62.139C23, (C4×S3)⋊1D6, (C2×C12)⋊5D6, C4○D12⋊8S3, (S3×D12)⋊7C2, C3⋊D4⋊12D6, (C6×D12)⋊17C2, (C2×D12)⋊14S3, C3⋊1(D4○D12), (C6×C12)⋊7C22, (C22×S3)⋊5D6, D12⋊5S3⋊8C2, D12⋊S3⋊8C2, D6⋊D6⋊10C2, C3⋊1(D4⋊6D6), (S3×C12)⋊2C22, (S3×C6).6C23, C6.14(S3×C23), (C3×C6).14C24, D6.7(C22×S3), C12.59D6⋊9C2, D6⋊S3⋊3C22, C3⋊D12⋊1C22, (C3×D12)⋊31C22, (S3×Dic3)⋊1C22, C32⋊7D4⋊7C22, C12⋊S3⋊23C22, C12.109(C22×S3), (C3×C12).119C23, (C3×Dic6)⋊30C22, C3⋊Dic3.18C23, (C3×Dic3).9C23, Dic3.6(C22×S3), C32⋊4Q8⋊22C22, (C2×C4)⋊3S32, C4.80(C2×S32), (C2×S32)⋊2C22, (S3×C3⋊D4)⋊1C2, (S3×C2×C6)⋊8C22, C22.8(C2×S32), (C4×C3⋊S3)⋊1C22, C2.16(C22×S32), (C3×C4○D12)⋊13C2, (C3×C3⋊D4)⋊8C22, (C2×C3⋊S3).20C23, (C2×C6).155(C22×S3), SmallGroup(288,955)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊24D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=cac-1=a-1, dad=a5, cbc-1=a4b, dbd=a10b, dcd=c-1 >
Subgroups: 1434 in 352 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×Dic3, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C2×S32, S3×C2×C6, D4⋊6D6, D4○D12, D12⋊5S3, D12⋊S3, S3×D12, D6⋊D6, S3×C3⋊D4, C6×D12, C3×C4○D12, C12.59D6, D12⋊24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4⋊6D6, D4○D12, C22×S32, D12⋊24D6
►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 13)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 38 5 46 9 42)(2 37 6 45 10 41)(3 48 7 44 11 40)(4 47 8 43 12 39)(13 28 21 32 17 36)(14 27 22 31 18 35)(15 26 23 30 19 34)(16 25 24 29 20 33)
(1 35)(2 28)(3 33)(4 26)(5 31)(6 36)(7 29)(8 34)(9 27)(10 32)(11 25)(12 30)(13 37)(14 42)(15 47)(16 40)(17 45)(18 38)(19 43)(20 48)(21 41)(22 46)(23 39)(24 44)
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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,38,5,46,9,42)(2,37,6,45,10,41)(3,48,7,44,11,40)(4,47,8,43,12,39)(13,28,21,32,17,36)(14,27,22,31,18,35)(15,26,23,30,19,34)(16,25,24,29,20,33), (1,35)(2,28)(3,33)(4,26)(5,31)(6,36)(7,29)(8,34)(9,27)(10,32)(11,25)(12,30)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44)>;
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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,38,5,46,9,42)(2,37,6,45,10,41)(3,48,7,44,11,40)(4,47,8,43,12,39)(13,28,21,32,17,36)(14,27,22,31,18,35)(15,26,23,30,19,34)(16,25,24,29,20,33), (1,35)(2,28)(3,33)(4,26)(5,31)(6,36)(7,29)(8,34)(9,27)(10,32)(11,25)(12,30)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44) );
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)], [(1,13),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,38,5,46,9,42),(2,37,6,45,10,41),(3,48,7,44,11,40),(4,47,8,43,12,39),(13,28,21,32,17,36),(14,27,22,31,18,35),(15,26,23,30,19,34),(16,25,24,29,20,33)], [(1,35),(2,28),(3,33),(4,26),(5,31),(6,36),(7,29),(8,34),(9,27),(10,32),(11,25),(12,30),(13,37),(14,42),(15,47),(16,40),(17,45),(18,38),(19,43),(20,48),(21,41),(22,46),(23,39),(24,44)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | ··· | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | ··· | 6N | 12A | 12B | 12C | ··· | 12I | 12J | 12K |
| order | 1 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | ··· | 6 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | C2×S32 | D4⋊6D6 | D4○D12 | D12⋊24D6 |
| kernel | D12⋊24D6 | D12⋊5S3 | D12⋊S3 | S3×D12 | D6⋊D6 | S3×C3⋊D4 | C6×D12 | C3×C4○D12 | C12.59D6 | C2×D12 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C22×S3 | C32 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of D12⋊24D6 ►in GL4(𝔽13) generated by
| 5 | 5 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 5 |
| 5 | 5 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(13))| [5,8,0,0,5,0,0,0,0,0,8,5,0,0,8,0],[0,0,5,0,0,0,5,8,8,0,0,0,8,5,0,0],[0,0,6,10,0,0,3,7,10,6,0,0,3,3,0,0],[6,10,0,0,3,7,0,0,0,0,10,6,0,0,3,3] >;
D12⋊24D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{24}D_6 % in TeX
G:=Group("D12:24D6"); // GroupNames label
G:=SmallGroup(288,955);
// by ID
G=gap.SmallGroup(288,955);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^4*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations