metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).12D6, C3⋊C8.30D4, C8⋊C22⋊2S3, D4⋊D6⋊4C2, C4○D4.40D6, (C3×D4).11D4, (C2×D4).79D6, C4.178(S3×D4), (C3×Q8).11D4, C12.D4⋊9C2, C12.195(C2×D4), D4.4(C3⋊D4), C3⋊5(D4.4D4), D4.Dic3⋊5C2, C12.46D4⋊6C2, C12.53D4⋊5C2, (C2×C12).14C23, Q8.11(C3⋊D4), C6.124(C4⋊D4), (C6×D4).104C22, (C2×D12).129C22, (C3×M4(2)).9C22, C4.Dic3.24C22, C2.30(C23.14D6), C22.13(D4⋊2S3), (C2×D4⋊S3)⋊22C2, (C3×C8⋊C22)⋊6C2, C4.51(C2×C3⋊D4), (C2×C6).36(C4○D4), (C2×C3⋊C8).170C22, (C2×C4).14(C22×S3), (C3×C4○D4).12C22, SmallGroup(192,758)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).D6
G = < a,b,c,d | a8=b2=c6=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 336 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C3⋊C8, C3⋊C8, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, Q8⋊2S3, C3×M4(2), C3×D8, C3×SD16, C2×D12, C6×D4, C3×C4○D4, D4.4D4, C12.53D4, C12.46D4, C12.D4, C2×D4⋊S3, D4.Dic3, D4⋊D6, C3×C8⋊C22, M4(2).D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.4D4, C23.14D6, M4(2).D6
Character table of M4(2).D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 8 | 24 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 6 | 6 | 8 | 12 | 12 | 12 | 24 | 4 | 4 | 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 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | 2 | -2 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | -2 | -1 | -1 | 1 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | -1 | 1 | √-3 | -√-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | -1 | 1 | -√-3 | √-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | -1 | 1 | √-3 | -√-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | -2 | -1 | 1 | -√-3 | √-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | -2 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | -2 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ26 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(1 35 46 22 11 29)(2 34 47 21 12 28)(3 33 48 20 13 27)(4 40 41 19 14 26)(5 39 42 18 15 25)(6 38 43 17 16 32)(7 37 44 24 9 31)(8 36 45 23 10 30)
(1 29 7 27 5 25 3 31)(2 28 8 26 6 32 4 30)(9 33 15 39 13 37 11 35)(10 40 16 38 14 36 12 34)(17 41 23 47 21 45 19 43)(18 48 24 46 22 44 20 42)
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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,35,46,22,11,29)(2,34,47,21,12,28)(3,33,48,20,13,27)(4,40,41,19,14,26)(5,39,42,18,15,25)(6,38,43,17,16,32)(7,37,44,24,9,31)(8,36,45,23,10,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,33,15,39,13,37,11,35)(10,40,16,38,14,36,12,34)(17,41,23,47,21,45,19,43)(18,48,24,46,22,44,20,42)>;
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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,35,46,22,11,29)(2,34,47,21,12,28)(3,33,48,20,13,27)(4,40,41,19,14,26)(5,39,42,18,15,25)(6,38,43,17,16,32)(7,37,44,24,9,31)(8,36,45,23,10,30), (1,29,7,27,5,25,3,31)(2,28,8,26,6,32,4,30)(9,33,15,39,13,37,11,35)(10,40,16,38,14,36,12,34)(17,41,23,47,21,45,19,43)(18,48,24,46,22,44,20,42) );
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)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48)], [(1,35,46,22,11,29),(2,34,47,21,12,28),(3,33,48,20,13,27),(4,40,41,19,14,26),(5,39,42,18,15,25),(6,38,43,17,16,32),(7,37,44,24,9,31),(8,36,45,23,10,30)], [(1,29,7,27,5,25,3,31),(2,28,8,26,6,32,4,30),(9,33,15,39,13,37,11,35),(10,40,16,38,14,36,12,34),(17,41,23,47,21,45,19,43),(18,48,24,46,22,44,20,42)]])
Matrix representation of M4(2).D6 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 21 | 0 | 32 |
| 0 | 0 | 40 | 21 | 41 | 32 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 15 | 33 | 21 | 12 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 39 | 15 | 0 | 1 |
| 36 | 28 | 0 | 0 | 0 | 0 |
| 28 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 58 | 1 | 71 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 46 | 15 |
| 37 | 28 | 0 | 0 | 0 | 0 |
| 45 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 58 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 31 | 1 | 46 | 39 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,72,0,0,0,0,0,0,0,40,40,16,15,0,0,21,21,16,33,0,0,0,41,0,21,0,0,32,32,0,12],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,39,0,0,0,72,0,15,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,0,34,1,1,0,0,0,58,0,39,0,0,1,1,0,46,0,0,0,71,0,15],[37,45,0,0,0,0,28,36,0,0,0,0,0,0,34,0,1,31,0,0,58,0,0,1,0,0,1,1,0,46,0,0,71,0,0,39] >;
M4(2).D6 in GAP, Magma, Sage, TeX
M_4(2).D_6
% in TeX
G:=Group("M4(2).D6"); // GroupNames label
G:=SmallGroup(192,758);
// by ID
G=gap.SmallGroup(192,758);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,1123,297,136,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations
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