metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1D4, Dic3⋊2D4, C23.10D6, (C2×C4).7D6, C2.9(S3×D4), D6⋊C4⋊11C2, (C2×D12)⋊3C2, C3⋊1(C4⋊D4), C22⋊C4⋊4S3, C6.20(C2×D4), Dic3⋊C4⋊5C2, C6.9(C4○D4), (C2×C6).25C23, C2.11(C4○D12), (C2×C12).53C22, (C22×C6).14C22, C22.43(C22×S3), (C2×Dic3).7C22, (C22×S3).18C22, (S3×C2×C4)⋊11C2, (C2×C3⋊D4)⋊2C2, (C3×C22⋊C4)⋊6C2, SmallGroup(96,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=cac-1=dad=a-1, cbc-1=a3b, bd=db, dcd=c-1 >
Subgroups: 250 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4⋊D4, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, Dic3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4, Dic3⋊D4
Character table of Dic3⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 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 | 2 | 0 | 0 | 0 | -1 | -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 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 2 | 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 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | i | -i | √3 | -√3 | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -i | i | √3 | -√3 | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | i | -i | -√3 | √3 | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -i | i | -√3 | √3 | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 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 28 4 25)(2 27 5 30)(3 26 6 29)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 23 16 20)(14 22 17 19)(15 21 18 24)(37 47 40 44)(38 46 41 43)(39 45 42 48)
(1 32 13 37)(2 31 14 42)(3 36 15 41)(4 35 16 40)(5 34 17 39)(6 33 18 38)(7 21 46 26)(8 20 47 25)(9 19 48 30)(10 24 43 29)(11 23 44 28)(12 22 45 27)
(1 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 47)(14 46)(15 45)(16 44)(17 43)(18 48)(19 38)(20 37)(21 42)(22 41)(23 40)(24 39)(25 32)(26 31)(27 36)(28 35)(29 34)(30 33)
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), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,23,16,20)(14,22,17,19)(15,21,18,24)(37,47,40,44)(38,46,41,43)(39,45,42,48), (1,32,13,37)(2,31,14,42)(3,36,15,41)(4,35,16,40)(5,34,17,39)(6,33,18,38)(7,21,46,26)(8,20,47,25)(9,19,48,30)(10,24,43,29)(11,23,44,28)(12,22,45,27), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,47)(14,46)(15,45)(16,44)(17,43)(18,48)(19,38)(20,37)(21,42)(22,41)(23,40)(24,39)(25,32)(26,31)(27,36)(28,35)(29,34)(30,33)>;
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), (1,28,4,25)(2,27,5,30)(3,26,6,29)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,23,16,20)(14,22,17,19)(15,21,18,24)(37,47,40,44)(38,46,41,43)(39,45,42,48), (1,32,13,37)(2,31,14,42)(3,36,15,41)(4,35,16,40)(5,34,17,39)(6,33,18,38)(7,21,46,26)(8,20,47,25)(9,19,48,30)(10,24,43,29)(11,23,44,28)(12,22,45,27), (1,8)(2,7)(3,12)(4,11)(5,10)(6,9)(13,47)(14,46)(15,45)(16,44)(17,43)(18,48)(19,38)(20,37)(21,42)(22,41)(23,40)(24,39)(25,32)(26,31)(27,36)(28,35)(29,34)(30,33) );
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)], [(1,28,4,25),(2,27,5,30),(3,26,6,29),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,23,16,20),(14,22,17,19),(15,21,18,24),(37,47,40,44),(38,46,41,43),(39,45,42,48)], [(1,32,13,37),(2,31,14,42),(3,36,15,41),(4,35,16,40),(5,34,17,39),(6,33,18,38),(7,21,46,26),(8,20,47,25),(9,19,48,30),(10,24,43,29),(11,23,44,28),(12,22,45,27)], [(1,8),(2,7),(3,12),(4,11),(5,10),(6,9),(13,47),(14,46),(15,45),(16,44),(17,43),(18,48),(19,38),(20,37),(21,42),(22,41),(23,40),(24,39),(25,32),(26,31),(27,36),(28,35),(29,34),(30,33)]])
Dic3⋊D4 is a maximal subgroup of
C24.38D6 C24.41D6 C42.93D6 C42.95D6 C42.97D6 C42.100D6 C42.104D6 C42⋊14D6 C42.228D6 D12⋊23D4 Dic6⋊24D4 C42.113D6 C42⋊19D6 C42.116D6 C24⋊7D6 C24⋊8D6 C24.45D6 C24.47D6 C6.322+ 1+4 Dic6⋊20D4 S3×C4⋊D4 C6.722- 1+4 D12⋊19D4 C6.402+ 1+4 C6.442+ 1+4 C6.482+ 1+4 C6.172- 1+4 D12⋊22D4 Dic6⋊22D4 C6.202- 1+4 C6.222- 1+4 C6.592+ 1+4 C6.1202+ 1+4 C6.1212+ 1+4 C6.822- 1+4 C4⋊C4⋊28D6 C6.612+ 1+4 C6.642+ 1+4 C6.662+ 1+4 C6.682+ 1+4 C6.692+ 1+4 C42.233D6 C42.138D6 C42⋊20D6 D12⋊10D4 Dic6⋊10D4 C42⋊22D6 C42.145D6 C42⋊25D6 C42.189D6 C42.161D6 C42.163D6 C42.164D6 C42⋊27D6 D18⋊D4 C62.55C23 D6⋊D12 Dic3⋊3D12 C62.82C23 C62.100C23 C62.113C23 C62.228C23 Dic15⋊2D4 D6⋊D20 D30⋊2D4 D30⋊12D4 D30⋊7D4 Dic15⋊4D4 D30⋊9D4
Dic3⋊D4 is a maximal quotient of
C6.(C4×Q8) C6.(C4⋊Q8) (C2×C4)⋊9D12 D6⋊(C4⋊C4) D6⋊C4⋊5C4 C6.C22≀C2 C6.(C4⋊D4) (C22×C4).37D6 D12.2D4 D12.3D4 D12.6D4 D12.7D4 Dic6⋊2D4 Dic6.D4 D6⋊D8 D6⋊SD16 C3⋊C8⋊1D4 C3⋊C8⋊D4 D12⋊3D4 D12.D4 Dic3⋊Q16 Dic6.11D4 D6⋊2SD16 C3⋊(C8⋊D4) D6⋊1Q16 C3⋊C8.D4 Dic3⋊SD16 D12.12D4 C24.17D6 C24.19D6 C24.23D6 C24.24D6 C24.25D6 C23⋊3D12 D18⋊D4 C62.55C23 D6⋊D12 Dic3⋊3D12 C62.82C23 C62.100C23 C62.113C23 C62.228C23 Dic15⋊2D4 D6⋊D20 D30⋊2D4 D30⋊12D4 D30⋊7D4 Dic15⋊4D4 D30⋊9D4
Matrix representation of Dic3⋊D4 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,6,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,3,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,11,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes D_4 % in TeX
G:=Group("Dic3:D4"); // GroupNames label
G:=SmallGroup(96,91);
// by ID
G=gap.SmallGroup(96,91);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,506,188,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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