metabelian, supersoluble, monomial
Aliases: Dic6.5D6, D4.8S32, C3⋊C8.6D6, D4.S3⋊3S3, C6.58(S3×D4), (C3×D4).10D6, C3⋊3(D4.D6), C3⋊Dic3.57D4, C12.8(C22×S3), (C3×C12).8C23, Dic3.D6⋊4C2, C12.31D6⋊3C2, C32⋊3Q16⋊12C2, C2.18(Dic3⋊D6), C12.D6.2C2, C32⋊10(C8.C22), (D4×C32).4C22, C32⋊4Q8.7C22, (C3×Dic6).11C22, C4.8(C2×S32), (C3×D4.S3)⋊2C2, (C2×C3⋊S3).22D4, (C3×C6).123(C2×D4), (C3×C3⋊C8).11C22, (C4×C3⋊S3).14C22, SmallGroup(288,579)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.D6
G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a9c-1 >
Subgroups: 570 in 140 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C3×C3⋊C8, C6.D6, C32⋊2Q8, C3×Dic6, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, D4.D6, C12.31D6, C32⋊3Q16, C3×D4.S3, Dic3.D6, C12.D6, Dic6.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S32, S3×D4, C2×S32, D4.D6, Dic3⋊D6, Dic6.D6
Character table of Dic6.D6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 4 | 18 | 2 | 2 | 4 | 2 | 12 | 12 | 18 | 36 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 4 | 4 | 8 | 24 | 24 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 1 | 1 | -2 | 1 | 0 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | 2 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -2 | 0 | -1 | 2 | -1 | 0 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | -2 | -2 | 0 | -1 | 2 | -1 | 0 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | 1 | 1 | -2 | 1 | 0 | 2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 2 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | -2 | 2 | 0 | -1 | 2 | -1 | 0 | 1 | -1 | -1 | 0 | 0 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 0 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -3 | 3 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 3 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Dic3⋊D6 |
| ρ21 | 4 | 4 | 4 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ22 | 4 | 4 | -4 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -1 | -1 | 2 | 2 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | 4 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 0 | 0 | 4 | -2 | -2 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | 0 | 0 | symplectic lifted from D4.D6, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | 0 | 0 | symplectic lifted from D4.D6, Schur index 2 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(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 32 7 26)(2 31 8 25)(3 30 9 36)(4 29 10 35)(5 28 11 34)(6 27 12 33)(13 45 19 39)(14 44 20 38)(15 43 21 37)(16 42 22 48)(17 41 23 47)(18 40 24 46)
(1 14 9 22 5 18)(2 21 10 17 6 13)(3 16 11 24 7 20)(4 23 12 19 8 15)(25 46 29 38 33 42)(26 41 30 45 34 37)(27 48 31 40 35 44)(28 43 32 47 36 39)
(1 37 4 40 7 43 10 46)(2 38 5 41 8 44 11 47)(3 39 6 42 9 45 12 48)(13 30 16 33 19 36 22 27)(14 31 17 34 20 25 23 28)(15 32 18 35 21 26 24 29)
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), (1,32,7,26)(2,31,8,25)(3,30,9,36)(4,29,10,35)(5,28,11,34)(6,27,12,33)(13,45,19,39)(14,44,20,38)(15,43,21,37)(16,42,22,48)(17,41,23,47)(18,40,24,46), (1,14,9,22,5,18)(2,21,10,17,6,13)(3,16,11,24,7,20)(4,23,12,19,8,15)(25,46,29,38,33,42)(26,41,30,45,34,37)(27,48,31,40,35,44)(28,43,32,47,36,39), (1,37,4,40,7,43,10,46)(2,38,5,41,8,44,11,47)(3,39,6,42,9,45,12,48)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29)>;
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), (1,32,7,26)(2,31,8,25)(3,30,9,36)(4,29,10,35)(5,28,11,34)(6,27,12,33)(13,45,19,39)(14,44,20,38)(15,43,21,37)(16,42,22,48)(17,41,23,47)(18,40,24,46), (1,14,9,22,5,18)(2,21,10,17,6,13)(3,16,11,24,7,20)(4,23,12,19,8,15)(25,46,29,38,33,42)(26,41,30,45,34,37)(27,48,31,40,35,44)(28,43,32,47,36,39), (1,37,4,40,7,43,10,46)(2,38,5,41,8,44,11,47)(3,39,6,42,9,45,12,48)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29) );
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)], [(1,32,7,26),(2,31,8,25),(3,30,9,36),(4,29,10,35),(5,28,11,34),(6,27,12,33),(13,45,19,39),(14,44,20,38),(15,43,21,37),(16,42,22,48),(17,41,23,47),(18,40,24,46)], [(1,14,9,22,5,18),(2,21,10,17,6,13),(3,16,11,24,7,20),(4,23,12,19,8,15),(25,46,29,38,33,42),(26,41,30,45,34,37),(27,48,31,40,35,44),(28,43,32,47,36,39)], [(1,37,4,40,7,43,10,46),(2,38,5,41,8,44,11,47),(3,39,6,42,9,45,12,48),(13,30,16,33,19,36,22,27),(14,31,17,34,20,25,23,28),(15,32,18,35,21,26,24,29)]])
Matrix representation of Dic6.D6 ►in GL8(𝔽73)
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 67 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 6 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 72 | 0 |
| 67 | 67 | 71 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 36 | 36 | 6 | 0 | 0 | 0 | 0 | 0 |
| 36 | 37 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 67 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 | 47 | 47 |
| 0 | 0 | 0 | 0 | 7 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 66 | 66 | 67 | 67 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 2 | 1 | 1 | 0 | 0 | 0 | 0 |
| 2 | 69 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 57 | 46 | 19 |
| 0 | 0 | 0 | 0 | 16 | 8 | 54 | 27 |
| 0 | 0 | 0 | 0 | 15 | 30 | 8 | 16 |
| 0 | 0 | 0 | 0 | 43 | 58 | 57 | 65 |
| 6 | 6 | 2 | 1 | 0 | 0 | 0 | 0 |
| 6 | 6 | 1 | 2 | 0 | 0 | 0 | 0 |
| 24 | 25 | 67 | 67 | 0 | 0 | 0 | 0 |
| 25 | 24 | 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 63 | 5 | 10 |
| 0 | 0 | 0 | 0 | 10 | 5 | 63 | 68 |
| 0 | 0 | 0 | 0 | 34 | 68 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 39 | 0 | 0 |
G:=sub<GL(8,GF(73))| [1,1,67,0,0,0,0,0,72,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,1,72,1,0,0,0,0,72,0,72,0,0,0,0,0,2,71,1,72,0,0,0,0,2,0,1,0],[67,0,36,36,0,0,0,0,67,0,36,37,0,0,0,0,71,72,6,6,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,67,6,7,66,0,0,0,0,0,6,0,66,0,0,0,0,26,47,6,67,0,0,0,0,0,47,0,67],[72,72,8,2,0,0,0,0,1,0,2,69,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,65,16,15,43,0,0,0,0,57,8,30,58,0,0,0,0,46,54,8,57,0,0,0,0,19,27,16,65],[6,6,24,25,0,0,0,0,6,6,25,24,0,0,0,0,2,1,67,67,0,0,0,0,1,2,67,67,0,0,0,0,0,0,0,0,68,10,34,5,0,0,0,0,63,5,68,39,0,0,0,0,5,63,0,0,0,0,0,0,10,68,0,0] >;
Dic6.D6 in GAP, Magma, Sage, TeX
{\rm Dic}_6.D_6 % in TeX
G:=Group("Dic6.D6"); // GroupNames label
G:=SmallGroup(288,579);
// by ID
G=gap.SmallGroup(288,579);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^-1>;
// generators/relations
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