metabelian, supersoluble, monomial
Aliases: D12⋊20D6, Dic6⋊18D6, C62.42D4, C4○D12⋊3S3, (C2×D12)⋊8S3, (C6×D12)⋊2C2, (C3×C12).61D4, C32⋊2D8⋊5C2, C3⋊4(D4⋊D6), (C2×C12).112D6, C32⋊6(C8⋊C22), Dic6⋊S3⋊5C2, C3⋊4(D12⋊6C22), C12.58D6⋊9C2, (C3×D12)⋊20C22, C12.43(C3⋊D4), (C3×C12).59C23, (C6×C12).72C22, C12.79(C22×S3), C32⋊4C8⋊2C22, C4.13(D6⋊S3), (C3×Dic6)⋊18C22, C22.4(D6⋊S3), (C2×C4).5S32, C4.74(C2×S32), (C3×C4○D12)⋊2C2, (C3×C6).63(C2×D4), C6.71(C2×C3⋊D4), C2.6(C2×D6⋊S3), (C2×C6).55(C3⋊D4), SmallGroup(288,471)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊20D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, ac=ca, dad=a7, cbc-1=a6b, dbd=a3b, dcd=c-1 >
Subgroups: 530 in 146 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C2×D12, C4○D12, C6×D4, C3×C4○D4, C32⋊4C8, C3×Dic6, S3×C12, C3×D12, C3×D12, C3×D12, C3×C3⋊D4, C6×C12, S3×C2×C6, D12⋊6C22, D4⋊D6, C32⋊2D8, Dic6⋊S3, C12.58D6, C6×D12, C3×C4○D12, D12⋊20D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, C2×C3⋊D4, D6⋊S3, C2×S32, D12⋊6C22, D4⋊D6, C2×D6⋊S3, D12⋊20D6
►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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)(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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 30)(2 25)(3 32)(4 27)(5 34)(6 29)(7 36)(8 31)(9 26)(10 33)(11 28)(12 35)(13 41)(14 48)(15 43)(16 38)(17 45)(18 40)(19 47)(20 42)(21 37)(22 44)(23 39)(24 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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,39,41,43,45,47)(38,40,42,44,46,48), (1,30)(2,25)(3,32)(4,27)(5,34)(6,29)(7,36)(8,31)(9,26)(10,33)(11,28)(12,35)(13,41)(14,48)(15,43)(16,38)(17,45)(18,40)(19,47)(20,42)(21,37)(22,44)(23,39)(24,46)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,39,41,43,45,47)(38,40,42,44,46,48), (1,30)(2,25)(3,32)(4,27)(5,34)(6,29)(7,36)(8,31)(9,26)(10,33)(11,28)(12,35)(13,41)(14,48)(15,43)(16,38)(17,45)(18,40)(19,47)(20,42)(21,37)(22,44)(23,39)(24,46) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,23,21,19,17,15),(14,24,22,20,18,16),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,39,41,43,45,47),(38,40,42,44,46,48)], [(1,30),(2,25),(3,32),(4,27),(5,34),(6,29),(7,36),(8,31),(9,26),(10,33),(11,28),(12,35),(13,41),(14,48),(15,43),(16,38),(17,45),(18,40),(19,47),(20,42),(21,37),(22,44),(23,39),(24,46)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | ··· | 6N | 8A | 8B | 12A | 12B | 12C | ··· | 12I | 12J | 12K |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 12 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 36 | 36 | 2 | 2 | 4 | ··· | 4 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S32 | D6⋊S3 | C2×S32 | D6⋊S3 | D12⋊6C22 | D4⋊D6 | D12⋊20D6 |
| kernel | D12⋊20D6 | C32⋊2D8 | Dic6⋊S3 | C12.58D6 | C6×D12 | C3×C4○D12 | C2×D12 | C4○D12 | C3×C12 | C62 | Dic6 | D12 | C2×C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D12⋊20D6 ►in GL4(𝔽73) generated by
| 49 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 70 |
| 0 | 3 | 0 | 0 |
| 49 | 0 | 0 | 0 |
| 0 | 0 | 0 | 65 |
| 0 | 0 | 9 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
G:=sub<GL(4,GF(73))| [49,0,0,0,0,3,0,0,0,0,24,0,0,0,0,70],[0,49,0,0,3,0,0,0,0,0,0,9,0,0,65,0],[8,0,0,0,0,65,0,0,0,0,64,0,0,0,0,9],[0,0,8,0,0,0,0,65,64,0,0,0,0,9,0,0] >;
D12⋊20D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{20}D_6 % in TeX
G:=Group("D12:20D6"); // GroupNames label
G:=SmallGroup(288,471);
// by ID
G=gap.SmallGroup(288,471);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c^-1=a^6*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations