metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.1D4, M4(2)⋊1D6, Dic6.1D4, C23.6D12, C4.78(S3×D4), (C2×D4).13D6, C12.91(C2×D4), C4.D4⋊1S3, C8.D6⋊5C2, D12⋊C4⋊1C2, C6.15C22≀C2, D4⋊6D6.2C2, C3⋊1(D4.9D4), (C2×C12).3C23, (C22×C6).19D4, C23.12D6⋊1C2, C4○D12.1C22, (C4×Dic3)⋊1C22, (C6×D4).13C22, C22.10(C2×D12), C2.18(D6⋊D4), (C2×Dic6)⋊12C22, (C3×M4(2))⋊8C22, (C2×C6).20(C2×D4), (C3×C4.D4)⋊3C2, (C2×C4).3(C22×S3), SmallGroup(192,305)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a5, cac-1=ab, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 528 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C24⋊C2, Dic12, C4×Dic3, C6.D4, C3×M4(2), C2×Dic6, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, D4.9D4, D12⋊C4, C3×C4.D4, C8.D6, C23.12D6, D4⋊6D6, M4(2)⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.9D4, D6⋊D4, M4(2)⋊D6
Character table of M4(2)⋊D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 24 | 2 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -2 | 2 | -1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 2 | -2 | -1 | -1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(33 47)(34 44)(35 41)(36 46)(37 43)(38 48)(39 45)(40 42)
(1 23 46 13 32 38)(2 31 47 8 25 45)(3 21 48 11 26 36)(4 29 41 6 27 43)(5 19 42 9 28 34)(7 17 44 15 30 40)(10 22 35 12 20 37)(14 18 39 16 24 33)
(1 38)(2 43)(3 40)(4 45)(5 34)(6 47)(7 36)(8 41)(9 42)(10 35)(11 44)(12 37)(13 46)(14 39)(15 48)(16 33)(17 26)(19 28)(21 30)(23 32)(25 29)(27 31)
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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42), (1,23,46,13,32,38)(2,31,47,8,25,45)(3,21,48,11,26,36)(4,29,41,6,27,43)(5,19,42,9,28,34)(7,17,44,15,30,40)(10,22,35,12,20,37)(14,18,39,16,24,33), (1,38)(2,43)(3,40)(4,45)(5,34)(6,47)(7,36)(8,41)(9,42)(10,35)(11,44)(12,37)(13,46)(14,39)(15,48)(16,33)(17,26)(19,28)(21,30)(23,32)(25,29)(27,31)>;
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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42), (1,23,46,13,32,38)(2,31,47,8,25,45)(3,21,48,11,26,36)(4,29,41,6,27,43)(5,19,42,9,28,34)(7,17,44,15,30,40)(10,22,35,12,20,37)(14,18,39,16,24,33), (1,38)(2,43)(3,40)(4,45)(5,34)(6,47)(7,36)(8,41)(9,42)(10,35)(11,44)(12,37)(13,46)(14,39)(15,48)(16,33)(17,26)(19,28)(21,30)(23,32)(25,29)(27,31) );
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)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,28),(18,25),(19,30),(20,27),(21,32),(22,29),(23,26),(24,31),(33,47),(34,44),(35,41),(36,46),(37,43),(38,48),(39,45),(40,42)], [(1,23,46,13,32,38),(2,31,47,8,25,45),(3,21,48,11,26,36),(4,29,41,6,27,43),(5,19,42,9,28,34),(7,17,44,15,30,40),(10,22,35,12,20,37),(14,18,39,16,24,33)], [(1,38),(2,43),(3,40),(4,45),(5,34),(6,47),(7,36),(8,41),(9,42),(10,35),(11,44),(12,37),(13,46),(14,39),(15,48),(16,33),(17,26),(19,28),(21,30),(23,32),(25,29),(27,31)]])
Matrix representation of M4(2)⋊D6 ►in GL8(𝔽73)
| 59 | 0 | 23 | 23 | 0 | 0 | 0 | 0 |
| 0 | 59 | 27 | 23 | 0 | 0 | 0 | 0 |
| 67 | 6 | 14 | 0 | 0 | 0 | 0 | 0 |
| 61 | 67 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 | 64 | 9 |
| 0 | 0 | 0 | 0 | 27 | 0 | 64 | 9 |
| 0 | 0 | 0 | 0 | 4 | 69 | 46 | 0 |
| 0 | 0 | 0 | 0 | 4 | 69 | 0 | 27 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 59 | 66 | 0 | 72 | 0 | 0 | 0 | 0 |
| 66 | 59 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 72 |
| 0 | 0 | 0 | 0 | 3 | 0 | 72 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 66 | 59 | 1 | 72 | 0 | 0 | 0 | 0 |
| 59 | 66 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 25 |
| 0 | 0 | 0 | 0 | 1 | 0 | 48 | 0 |
| 0 | 0 | 0 | 0 | 0 | 70 | 0 | 1 |
| 0 | 0 | 0 | 0 | 3 | 0 | 72 | 0 |
G:=sub<GL(8,GF(73))| [59,0,67,61,0,0,0,0,0,59,6,67,0,0,0,0,23,27,14,0,0,0,0,0,23,23,0,14,0,0,0,0,0,0,0,0,0,27,4,4,0,0,0,0,46,0,69,69,0,0,0,0,64,64,46,0,0,0,0,0,9,9,0,27],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,72,59,66,0,0,0,0,1,0,66,59,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0],[1,0,66,59,0,0,0,0,1,72,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,72,0,70,0,0,0,0,0,0,48,0,72,0,0,0,0,25,0,1,0] >;
M4(2)⋊D6 in GAP, Magma, Sage, TeX
M_4(2)\rtimes D_6
% in TeX
G:=Group("M4(2):D6"); // GroupNames label
G:=SmallGroup(192,305);
// by ID
G=gap.SmallGroup(192,305);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,1123,570,136,1684,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
Export