metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.5D4, C4⋊C4⋊2S3, D6⋊C4⋊13C2, C2.12(S3×D4), (C2×C4).11D6, C6.25(C2×D4), Dic3⋊C4⋊6C2, (C2×D12).3C2, C6.12(C4○D4), (C2×C6).35C23, C2.14(C4○D12), (C2×C12).57C22, C2.5(Q8⋊3S3), C3⋊3(C22.D4), (C22×S3).6C22, C22.49(C22×S3), (C2×Dic3).11C22, (C3×C4⋊C4)⋊5C2, (S3×C2×C4)⋊13C2, SmallGroup(96,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.D4
G = < a,b,c,d | a6=b2=c4=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 202 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C22.D4, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, D6.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D12, S3×D4, Q8⋊3S3, D6.D4
Character table of D6.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2i | 0 | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2i | 0 | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | -√-3 | i | √-3 | -√3 | √3 | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -√-3 | -i | √-3 | √3 | -√3 | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | √-3 | -i | -√-3 | -√3 | √3 | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | √-3 | i | -√-3 | √3 | -√3 | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
►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 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)
(1 29 17 24)(2 30 18 19)(3 25 13 20)(4 26 14 21)(5 27 15 22)(6 28 16 23)(7 34 46 39)(8 35 47 40)(9 36 48 41)(10 31 43 42)(11 32 44 37)(12 33 45 38)
(1 12 4 9)(2 7 5 10)(3 8 6 11)(13 47 16 44)(14 48 17 45)(15 43 18 46)(19 34 22 31)(20 35 23 32)(21 36 24 33)(25 40 28 37)(26 41 29 38)(27 42 30 39)
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), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,47,16,44)(14,48,17,45)(15,43,18,46)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39)>;
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), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31), (1,29,17,24)(2,30,18,19)(3,25,13,20)(4,26,14,21)(5,27,15,22)(6,28,16,23)(7,34,46,39)(8,35,47,40)(9,36,48,41)(10,31,43,42)(11,32,44,37)(12,33,45,38), (1,12,4,9)(2,7,5,10)(3,8,6,11)(13,47,16,44)(14,48,17,45)(15,43,18,46)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39) );
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)], [(1,8),(2,7),(3,12),(4,11),(5,10),(6,9),(13,45),(14,44),(15,43),(16,48),(17,47),(18,46),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31)], [(1,29,17,24),(2,30,18,19),(3,25,13,20),(4,26,14,21),(5,27,15,22),(6,28,16,23),(7,34,46,39),(8,35,47,40),(9,36,48,41),(10,31,43,42),(11,32,44,37),(12,33,45,38)], [(1,12,4,9),(2,7,5,10),(3,8,6,11),(13,47,16,44),(14,48,17,45),(15,43,18,46),(19,34,22,31),(20,35,23,32),(21,36,24,33),(25,40,28,37),(26,41,29,38),(27,42,30,39)]])
D6.D4 is a maximal subgroup of
C6.2- 1+4 C6.112+ 1+4 C6.62- 1+4 C42⋊12D6 C42.93D6 C42.100D6 C42.104D6 C42⋊14D6 D12⋊23D4 C42⋊19D6 C42.131D6 C42.132D6 C42.133D6 C42.136D6 C6.372+ 1+4 C6.402+ 1+4 C6.442+ 1+4 C6.482+ 1+4 C4⋊C4⋊26D6 C6.162- 1+4 D12⋊21D4 D12⋊22D4 C6.532+ 1+4 C6.202- 1+4 C6.222- 1+4 C6.562+ 1+4 C6.782- 1+4 C6.592+ 1+4 S3×C22.D4 C6.1202+ 1+4 C6.1212+ 1+4 C6.822- 1+4 C6.612+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C6.692+ 1+4 C42.237D6 C42.150D6 C42.151D6 C42.152D6 C42.153D6 C42.155D6 C42.157D6 C42.158D6 C42⋊25D6 C42⋊26D6 C42.189D6 C42.161D6 C42.163D6 C42.164D6 C42⋊27D6 C42.171D6 D12⋊12D4 C42.178D6 C42.180D6 D18.D4 C62.20C23 C62.23C23 C62.24C23 C62.54C23 D6.D12 C62.67C23 C62.238C23 D6⋊Dic5⋊C2 D30.35D4 (C2×D12).D5 D6.D20 D30.6D4 D30.7D4 D30.29D4
D6.D4 is a maximal quotient of
Dic3⋊C4⋊C4 (C2×C4).Dic6 C22.58(S3×D4) (C2×C4)⋊9D12 D6⋊C4⋊5C4 C6.C22≀C2 C6.(C4⋊D4) D6.2SD16 D6.4SD16 C4.Q8⋊S3 C6.(C4○D8) D6.5D8 D6.2Q16 C2.D8⋊S3 C2.D8⋊7S3 C4⋊C4⋊5Dic3 (C2×C12).54D4 D6⋊C4⋊6C4 D6⋊C4⋊7C4 (C2×C4)⋊3D12 (C2×C12).289D4 (C2×C12).290D4 D18.D4 C62.20C23 C62.23C23 C62.24C23 C62.54C23 D6.D12 C62.67C23 C62.238C23 D6⋊Dic5⋊C2 D30.35D4 (C2×D12).D5 D6.D20 D30.6D4 D30.7D4 D30.29D4
Matrix representation of D6.D4 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 3 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 4 | 0 | 0 |
| 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 3 | 7 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[3,6,0,0,3,10,0,0,0,0,12,0,0,0,0,12],[11,9,0,0,4,2,0,0,0,0,0,12,0,0,1,0],[3,6,0,0,7,10,0,0,0,0,12,0,0,0,0,1] >;
D6.D4 in GAP, Magma, Sage, TeX
D_6.D_4
% in TeX
G:=Group("D6.D4"); // GroupNames label
G:=SmallGroup(96,101);
// by ID
G=gap.SmallGroup(96,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^4=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations
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