metabelian, supersoluble, monomial
Aliases: D12⋊12D6, C62.4C23, C32⋊32+ 1+4, D4⋊7S32, (C4×S3)⋊3D6, (S3×D4)⋊4S3, C3⋊D4⋊4D6, (C3×D4)⋊10D6, (C22×S3)⋊6D6, D6⋊D6⋊12C2, C3⋊2(D4⋊6D6), (S3×C12)⋊7C22, D6.4D6⋊6C2, D6.D6⋊6C2, C6.20(S3×C23), (C3×C6).20C24, D12⋊5S3⋊10C2, C12.D6⋊8C2, (C3×D12)⋊14C22, (S3×C6).11C23, (C3×C12).32C23, C12.32(C22×S3), (S3×Dic3)⋊2C22, D6.21(C22×S3), C32⋊7D4⋊4C22, D6⋊S3⋊14C22, C3⋊D12⋊15C22, C32⋊2Q8⋊13C22, (D4×C32)⋊12C22, C3⋊Dic3.22C23, C32⋊4Q8⋊10C22, Dic3.10(C22×S3), (C3×Dic3).14C23, (C3×S3×D4)⋊9C2, C4.32(C2×S32), (C2×S32)⋊4C22, (S3×C3⋊D4)⋊4C2, C22.4(C2×S32), (C4×C3⋊S3)⋊4C22, (S3×C2×C6)⋊10C22, C2.22(C22×S32), (C3×C3⋊D4)⋊4C22, (C2×C6).5(C22×S3), (C2×D6⋊S3)⋊15C2, (C2×C3⋊S3).25C23, (C2×C3⋊Dic3)⋊11C22, SmallGroup(288,961)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊12D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 1338 in 352 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, 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, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, 2+ 1+4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, S3×D4, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, S3×Dic3, D6⋊S3, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C2×S32, S3×C2×C6, D4⋊6D6, D12⋊5S3, D6.D6, D6⋊D6, D6.4D6, C2×D6⋊S3, S3×C3⋊D4, C3×S3×D4, C12.D6, D12⋊12D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4⋊6D6, C22×S32, D12⋊12D6
►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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)
(1 27 5 31 9 35)(2 34 6 26 10 30)(3 29 7 33 11 25)(4 36 8 28 12 32)(13 48 21 44 17 40)(14 43 22 39 18 47)(15 38 23 46 19 42)(16 45 24 41 20 37)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
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), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48), (1,27,5,31,9,35)(2,34,6,26,10,30)(3,29,7,33,11,25)(4,36,8,28,12,32)(13,48,21,44,17,40)(14,43,22,39,18,47)(15,38,23,46,19,42)(16,45,24,41,20,37), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)>;
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), (1,16)(2,15)(3,14)(4,13)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,48), (1,27,5,31,9,35)(2,34,6,26,10,30)(3,29,7,33,11,25)(4,36,8,28,12,32)(13,48,21,44,17,40)(14,43,22,39,18,47)(15,38,23,46,19,42)(16,45,24,41,20,37), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30) );
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)], [(1,16),(2,15),(3,14),(4,13),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(36,48)], [(1,27,5,31,9,35),(2,34,6,26,10,30),(3,29,7,33,11,25),(4,36,8,28,12,32),(13,48,21,44,17,40),(14,43,22,39,18,47),(15,38,23,46,19,42),(16,45,24,41,20,37)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 6Q | 12A | 12B | 12C | 12D | 12E |
| 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 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 18 | 2 | 2 | 4 | 2 | 6 | 6 | 18 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | C2×S32 | D4⋊6D6 | D12⋊12D6 |
| kernel | D12⋊12D6 | D12⋊5S3 | D6.D6 | D6⋊D6 | D6.4D6 | C2×D6⋊S3 | S3×C3⋊D4 | C3×S3×D4 | C12.D6 | S3×D4 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C22×S3 | C32 | D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 4 | 1 |
Matrix representation of D12⋊12D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 12 | 6 |
| 0 | 0 | 2 | 11 | 7 | 6 |
| 0 | 0 | 6 | 3 | 4 | 2 |
| 0 | 0 | 10 | 3 | 11 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 | 1 | 0 |
| 0 | 0 | 0 | 8 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,6,10,0,0,11,11,3,3,0,0,12,7,4,11,0,0,6,6,2,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,8,0,0,0,0,12,0,8,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;
D12⋊12D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{12}D_6 % in TeX
G:=Group("D12:12D6"); // GroupNames label
G:=SmallGroup(288,961);
// by ID
G=gap.SmallGroup(288,961);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations