Aliases: Q8.2D6, C22.2S4, GL2(𝔽3)⋊1C2, CSU2(𝔽3)⋊1C2, SL2(𝔽3).2C22, C2.7(C2×S4), (C2×Q8)⋊2S3, (C2×SL2(𝔽3))⋊3C2, SmallGroup(96,190)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — Q8.D6 |
Generators and relations for Q8.D6
G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=dad-1=a-1, cac-1=ab, cbc-1=a, dbd-1=a-1b, dcd-1=a2c-1 >
Character table of Q8.D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | |
| size | 1 | 1 | 2 | 12 | 8 | 6 | 6 | 12 | 8 | 8 | 8 | 12 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 3 | 3 | 3 | -1 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -3 | 1 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ9 | 3 | 3 | -3 | -1 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ10 | 3 | 3 | 3 | 1 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ11 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ12 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | -√-3 | -1 | √-3 | 0 | 0 | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | √-3 | -1 | -√-3 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T187
(1 10 3 15)(2 7 4 12)(5 14 16 9)(6 8 11 13)
(1 8 3 13)(2 5 4 16)(6 15 11 10)(7 9 12 14)
(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(1 4 3 2)(5 11 16 6)(7 15 12 10)(8 9 13 14)
G:=sub<Sym(16)| (1,10,3,15)(2,7,4,12)(5,14,16,9)(6,8,11,13), (1,8,3,13)(2,5,4,16)(6,15,11,10)(7,9,12,14), (1,2)(3,4)(5,6,7,8,9,10)(11,12,13,14,15,16), (1,4,3,2)(5,11,16,6)(7,15,12,10)(8,9,13,14)>;
G:=Group( (1,10,3,15)(2,7,4,12)(5,14,16,9)(6,8,11,13), (1,8,3,13)(2,5,4,16)(6,15,11,10)(7,9,12,14), (1,2)(3,4)(5,6,7,8,9,10)(11,12,13,14,15,16), (1,4,3,2)(5,11,16,6)(7,15,12,10)(8,9,13,14) );
G=PermutationGroup([[(1,10,3,15),(2,7,4,12),(5,14,16,9),(6,8,11,13)], [(1,8,3,13),(2,5,4,16),(6,15,11,10),(7,9,12,14)], [(1,2),(3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)], [(1,4,3,2),(5,11,16,6),(7,15,12,10),(8,9,13,14)]])
G:=TransitiveGroup(16,187);
►On 16 points - transitive group 16T192
(1 5 2 9)(3 16 4 13)(6 8 10 7)(11 12 14 15)
(1 6 2 10)(3 14 4 11)(5 7 9 8)(12 13 15 16)
(3 4)(5 6 7)(8 9 10)(11 12 13 14 15 16)
(1 3 2 4)(5 13 9 16)(6 15 10 12)(7 11 8 14)
G:=sub<Sym(16)| (1,5,2,9)(3,16,4,13)(6,8,10,7)(11,12,14,15), (1,6,2,10)(3,14,4,11)(5,7,9,8)(12,13,15,16), (3,4)(5,6,7)(8,9,10)(11,12,13,14,15,16), (1,3,2,4)(5,13,9,16)(6,15,10,12)(7,11,8,14)>;
G:=Group( (1,5,2,9)(3,16,4,13)(6,8,10,7)(11,12,14,15), (1,6,2,10)(3,14,4,11)(5,7,9,8)(12,13,15,16), (3,4)(5,6,7)(8,9,10)(11,12,13,14,15,16), (1,3,2,4)(5,13,9,16)(6,15,10,12)(7,11,8,14) );
G=PermutationGroup([[(1,5,2,9),(3,16,4,13),(6,8,10,7),(11,12,14,15)], [(1,6,2,10),(3,14,4,11),(5,7,9,8),(12,13,15,16)], [(3,4),(5,6,7),(8,9,10),(11,12,13,14,15,16)], [(1,3,2,4),(5,13,9,16),(6,15,10,12),(7,11,8,14)]])
G:=TransitiveGroup(16,192);
Q8.D6 is a maximal subgroup of
GL2(𝔽3)⋊C22 D4.4S4 D4.5S4 D6.S4 D6.2S4 SL2(𝔽3).D6 C22.S5 D10.1S4 D10.2S4 Q8.D30
Q8.D6 is a maximal quotient of
CSU2(𝔽3)⋊C4 Q8.Dic6 GL2(𝔽3)⋊C4 Q8.2D12 C23.14S4 C23.15S4 C23.16S4 Q8.D18 D6.S4 D6.2S4 SL2(𝔽3).D6 D10.1S4 D10.2S4 Q8.D30
Matrix representation of Q8.D6 ►in GL4(𝔽3) generated by
| 2 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 0 | 0 | 1 |
| 1 | 0 | 0 | 2 |
| 0 | 2 | 2 | 0 |
| 0 | 2 | 1 | 0 |
| 2 | 0 | 0 | 2 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(3))| [2,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1],[1,0,0,2,0,2,2,0,0,2,1,0,2,0,0,2],[1,0,0,0,0,0,1,0,0,2,1,0,1,0,0,1],[0,1,1,0,0,0,0,2,2,0,0,1,0,1,0,0] >;
Q8.D6 in GAP, Magma, Sage, TeX
Q_8.D_6
% in TeX
G:=Group("Q8.D6"); // GroupNames label
G:=SmallGroup(96,190);
// by ID
G=gap.SmallGroup(96,190);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,146,579,447,117,364,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a*b,c*b*c^-1=a,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
Export
Subgroup lattice of Q8.D6 in TeX
Character table of Q8.D6 in TeX