metabelian, supersoluble, monomial
Aliases: D6.Dic3, C12.30D6, Dic3.Dic3, C32⋊2M4(2), C3⋊C8⋊4S3, C4.15S32, (C4×S3).2S3, (S3×C6).2C4, C6.17(C4×S3), C3⋊3(C8⋊S3), (S3×C12).1C2, C32⋊4C8⋊7C2, C2.3(S3×Dic3), C6.2(C2×Dic3), C3⋊1(C4.Dic3), (C3×Dic3).1C4, (C3×C12).29C22, (C3×C3⋊C8)⋊7C2, (C3×C6).10(C2×C4), SmallGroup(144,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.Dic3
G = < a,b,c,d | a6=b2=1, c6=a3, d2=a3c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >
Character table of D6.Dic3
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 2 | 2 | 4 | 1 | 1 | 6 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | -i | i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | -1 | -1 | 2 | 2 | -2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 2 | -1 | -1 | -2 | -2 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 0 | -1 | 2 | -1 | -2 | -2 | 0 | -1 | 2 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | 0 | -1 | 2 | -1 | -2 | -2 | 0 | -1 | 2 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ17 | 2 | -2 | 0 | 2 | 2 | 2 | 2i | -2i | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | 0 | 2 | 2 | 2 | -2i | 2i | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | 0 | 2 | -1 | -1 | 2i | -2i | 0 | -2 | 1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -i | 2i | -2i | i | -i | i | -√3 | √3 | 0 | 0 | 0 | 0 | complex lifted from C4.Dic3 |
| ρ20 | 2 | -2 | 0 | 2 | -1 | -1 | -2i | 2i | 0 | -2 | 1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | i | -2i | 2i | -i | i | -i | -√3 | √3 | 0 | 0 | 0 | 0 | complex lifted from C4.Dic3 |
| ρ21 | 2 | -2 | 0 | 2 | -1 | -1 | -2i | 2i | 0 | -2 | 1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | i | -2i | 2i | -i | i | -i | √3 | -√3 | 0 | 0 | 0 | 0 | complex lifted from C4.Dic3 |
| ρ22 | 2 | -2 | 0 | 2 | -1 | -1 | 2i | -2i | 0 | -2 | 1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -i | 2i | -2i | i | -i | i | √3 | -√3 | 0 | 0 | 0 | 0 | complex lifted from C4.Dic3 |
| ρ23 | 2 | -2 | 0 | -1 | 2 | -1 | 2i | -2i | 0 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -i | i | -2i | -i | i | 0 | 0 | 2ζ8ζ3+ζ8 | 2ζ83ζ3+ζ83 | 2ζ87ζ3+ζ87 | 2ζ85ζ3+ζ85 | complex lifted from C8⋊S3 |
| ρ24 | 2 | -2 | 0 | -1 | 2 | -1 | -2i | 2i | 0 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | i | -i | 2i | i | -i | 0 | 0 | 2ζ83ζ3+ζ83 | 2ζ8ζ3+ζ8 | 2ζ85ζ3+ζ85 | 2ζ87ζ3+ζ87 | complex lifted from C8⋊S3 |
| ρ25 | 2 | -2 | 0 | -1 | 2 | -1 | 2i | -2i | 0 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -i | i | -2i | -i | i | 0 | 0 | 2ζ85ζ3+ζ85 | 2ζ87ζ3+ζ87 | 2ζ83ζ3+ζ83 | 2ζ8ζ3+ζ8 | complex lifted from C8⋊S3 |
| ρ26 | 2 | -2 | 0 | -1 | 2 | -1 | -2i | 2i | 0 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | i | -i | 2i | i | -i | 0 | 0 | 2ζ87ζ3+ζ87 | 2ζ85ζ3+ζ85 | 2ζ8ζ3+ζ8 | 2ζ83ζ3+ζ83 | complex lifted from C8⋊S3 |
| ρ27 | 4 | 4 | 0 | -2 | -2 | 1 | 4 | 4 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ28 | 4 | 4 | 0 | -2 | -2 | 1 | -4 | -4 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
| ρ29 | 4 | -4 | 0 | -2 | -2 | 1 | 4i | -4i | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | -2 | -2 | 1 | -4i | 4i | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 48 46 44 42 40)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)
(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 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 41 34 38 31 47 28 44)(26 46 35 43 32 40 29 37)(27 39 36 48 33 45 30 42)
G:=sub<Sym(48)| (1,11,9,7,5,3)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,16,18,20,22,24)(25,27,29,31,33,35)(26,28,30,32,34,36)(37,47,45,43,41,39)(38,48,46,44,42,40), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,41,34,38,31,47,28,44)(26,46,35,43,32,40,29,37)(27,39,36,48,33,45,30,42)>;
G:=Group( (1,11,9,7,5,3)(2,12,10,8,6,4)(13,15,17,19,21,23)(14,16,18,20,22,24)(25,27,29,31,33,35)(26,28,30,32,34,36)(37,47,45,43,41,39)(38,48,46,44,42,40), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,41,34,38,31,47,28,44)(26,46,35,43,32,40,29,37)(27,39,36,48,33,45,30,42) );
G=PermutationGroup([[(1,11,9,7,5,3),(2,12,10,8,6,4),(13,15,17,19,21,23),(14,16,18,20,22,24),(25,27,29,31,33,35),(26,28,30,32,34,36),(37,47,45,43,41,39),(38,48,46,44,42,40)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,37),(23,38),(24,39)], [(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)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,41,34,38,31,47,28,44),(26,46,35,43,32,40,29,37),(27,39,36,48,33,45,30,42)]])
D6.Dic3 is a maximal subgroup of
S3×C8⋊S3 C24⋊D6 C24.63D6 C24.64D6 S3×C4.Dic3 D12.2Dic3 D12.Dic3 Dic6⋊3D6 Dic6.19D6 D12⋊9D6 D12.7D6 D12⋊6D6 D12.11D6 D12.24D6 Dic6.22D6 C36.39D6 D6.Dic9 He3⋊M4(2) He3⋊3M4(2) C33⋊7M4(2) C33⋊8M4(2) C33⋊10M4(2)
D6.Dic3 is a maximal quotient of
C3⋊C8⋊Dic3 C12.77D12 C12.81D12 C36.39D6 D6.Dic9 He3⋊M4(2) C33⋊7M4(2) C33⋊8M4(2) C33⋊10M4(2)
Matrix representation of D6.Dic3 ►in GL4(𝔽5) generated by
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 2 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 2 |
| 0 | 1 | 0 | 4 |
| 4 | 0 | 1 | 0 |
| 0 | 2 | 0 | 4 |
| 3 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 |
| 0 | 1 | 0 | 4 |
| 3 | 0 | 1 | 0 |
| 0 | 2 | 0 | 2 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [4,0,1,0,0,4,0,1,2,0,2,0,0,2,0,2],[0,4,0,3,1,0,2,0,0,1,0,1,4,0,4,0],[2,0,3,0,0,1,0,2,1,0,1,0,0,4,0,2],[0,0,0,2,1,0,3,0,0,4,0,1,1,0,0,0] >;
D6.Dic3 in GAP, Magma, Sage, TeX
D_6.{\rm Dic}_3 % in TeX
G:=Group("D6.Dic3"); // GroupNames label
G:=SmallGroup(144,54);
// by ID
G=gap.SmallGroup(144,54);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^6=a^3,d^2=a^3*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of D6.Dic3 in TeX
Character table of D6.Dic3 in TeX