metabelian, supersoluble, monomial
Aliases: D12.2D6, C24.39D6, Dic6.2D6, C8.18S32, C24⋊C2⋊6S3, C6.32(S3×D4), C32⋊6(C4○D8), C32⋊2D8⋊4C2, D12⋊S3⋊2C2, C3⋊Dic3.42D4, C32⋊2Q16⋊3C2, C2.9(D6⋊D6), C3⋊2(Q8.7D6), C12.74(C22×S3), (C3×C24).37C22, (C3×C12).54C23, (C3×D12).7C22, (C3×Dic6).7C22, C32⋊4C8.22C22, (C8×C3⋊S3)⋊7C2, C4.69(C2×S32), (C2×C3⋊S3).43D4, (C3×C6).38(C2×D4), (C3×C24⋊C2)⋊12C2, (C4×C3⋊S3).68C22, SmallGroup(288,457)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.2D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=cac-1=a-1, dad-1=a5, cbc-1=ab, dbd-1=a4b, dcd-1=c5 >
Subgroups: 562 in 135 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C32⋊4C8, C3×C24, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C4×C3⋊S3, Q8.7D6, C32⋊2D8, C32⋊2Q16, C3×C24⋊C2, C8×C3⋊S3, D12⋊S3, D12.2D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S32, S3×D4, C2×S32, Q8.7D6, D6⋊D6, D12.2D6
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 36)(21 35)(22 34)(23 33)(24 32)
(1 26 11 28 9 30 7 32 5 34 3 36)(2 25 12 27 10 29 8 31 6 33 4 35)(13 42 23 44 21 46 19 48 17 38 15 40)(14 41 24 43 22 45 20 47 18 37 16 39)
(1 8 7 2)(3 6 9 12)(4 11 10 5)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 42 43 48)(38 47 44 41)(39 40 45 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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,36)(21,35)(22,34)(23,33)(24,32), (1,26,11,28,9,30,7,32,5,34,3,36)(2,25,12,27,10,29,8,31,6,33,4,35)(13,42,23,44,21,46,19,48,17,38,15,40)(14,41,24,43,22,45,20,47,18,37,16,39), (1,8,7,2)(3,6,9,12)(4,11,10,5)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,42,43,48)(38,47,44,41)(39,40,45,46)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,36)(21,35)(22,34)(23,33)(24,32), (1,26,11,28,9,30,7,32,5,34,3,36)(2,25,12,27,10,29,8,31,6,33,4,35)(13,42,23,44,21,46,19,48,17,38,15,40)(14,41,24,43,22,45,20,47,18,37,16,39), (1,8,7,2)(3,6,9,12)(4,11,10,5)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,42,43,48)(38,47,44,41)(39,40,45,46) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,36),(21,35),(22,34),(23,33),(24,32)], [(1,26,11,28,9,30,7,32,5,34,3,36),(2,25,12,27,10,29,8,31,6,33,4,35),(13,42,23,44,21,46,19,48,17,38,15,40),(14,41,24,43,22,45,20,47,18,37,16,39)], [(1,8,7,2),(3,6,9,12),(4,11,10,5),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,42,43,48),(38,47,44,41),(39,40,45,46)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 12 | 18 | 2 | 2 | 4 | 2 | 9 | 9 | 12 | 12 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D8 | S32 | S3×D4 | C2×S32 | Q8.7D6 | D6⋊D6 | D12.2D6 |
| kernel | D12.2D6 | C32⋊2D8 | C32⋊2Q16 | C3×C24⋊C2 | C8×C3⋊S3 | D12⋊S3 | C24⋊C2 | C3⋊Dic3 | C2×C3⋊S3 | C24 | Dic6 | D12 | C32 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 4 |
Matrix representation of D12.2D6 ►in GL6(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 22 | 0 | 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 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 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 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 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))| [46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,22,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
D12.2D6 in GAP, Magma, Sage, TeX
D_{12}._2D_6 % in TeX
G:=Group("D12.2D6"); // GroupNames label
G:=SmallGroup(288,457);
// by ID
G=gap.SmallGroup(288,457);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,58,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^5,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^5>;
// generators/relations