Aliases: Q8:D12, C4:GL2(F3), SL2(F3):1D4, (C4xQ8):2S3, C2.4(C4:S4), (C2xC4).11S4, (C2xQ8).9D6, C22.36(C2xS4), C2.4(C4.3S4), (C4xSL2(F3)):6C2, (C2xGL2(F3)):4C2, C2.4(C2xGL2(F3)), (C2xSL2(F3)).9C22, SmallGroup(192,952)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2xSL2(F3) — Q8:D12 |
| C1 — C2 — Q8 — SL2(F3) — C2xSL2(F3) — C2xGL2(F3) — Q8:D12 |
| SL2(F3) — C2xSL2(F3) — Q8:D12 |
Generators and relations for Q8:D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, cac-1=ab, cbc-1=a, dbd=a-1b, dcd=c-1 >
Subgroups: 473 in 91 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, C12, D6, C2xC6, C42, C4:C4, C2xC8, SD16, C2xD4, C2xQ8, SL2(F3), D12, C2xC12, C22xS3, D4:C4, C4:C8, C4xQ8, C4:1D4, C2xSD16, GL2(F3), C2xSL2(F3), C2xD12, C4:SD16, C4xSL2(F3), C2xGL2(F3), Q8:D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, GL2(F3), C2xS4, C4:S4, C2xGL2(F3), C4.3S4, Q8:D12
Character table of Q8:D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 8 | 2 | 2 | 6 | 6 | 12 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | -2 | 2 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | 1 | -1 | √-2 | √-2 | -√-2 | -√-2 | 1 | -1 | 1 | -1 | complex lifted from GL2(F3) |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | -√-2 | √-2 | √-2 | -√-2 | -1 | 1 | -1 | 1 | complex lifted from GL2(F3) |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | 1 | -1 | -√-2 | -√-2 | √-2 | √-2 | 1 | -1 | 1 | -1 | complex lifted from GL2(F3) |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | √-2 | -√-2 | -√-2 | √-2 | -1 | 1 | -1 | 1 | complex lifted from GL2(F3) |
| ρ14 | 3 | 3 | 3 | 3 | -1 | 1 | 0 | -3 | -3 | -1 | -1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 4 | -4 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from GL2(F3) |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | -4 | 4 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from GL2(F3) |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ23 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4:S4 |
►On 32 points
Generators in S32
(1 13 5 22)(2 10 6 31)(3 19 7 28)(4 16 8 25)(9 17 30 26)(11 24 32 15)(12 20 21 29)(14 27 23 18)
(1 17 5 26)(2 14 6 23)(3 11 7 32)(4 20 8 29)(9 22 30 13)(10 18 31 27)(12 25 21 16)(15 28 24 19)
(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)
(1 3)(5 7)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(32)| (1,13,5,22)(2,10,6,31)(3,19,7,28)(4,16,8,25)(9,17,30,26)(11,24,32,15)(12,20,21,29)(14,27,23,18), (1,17,5,26)(2,14,6,23)(3,11,7,32)(4,20,8,29)(9,22,30,13)(10,18,31,27)(12,25,21,16)(15,28,24,19), (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), (1,3)(5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (1,13,5,22)(2,10,6,31)(3,19,7,28)(4,16,8,25)(9,17,30,26)(11,24,32,15)(12,20,21,29)(14,27,23,18), (1,17,5,26)(2,14,6,23)(3,11,7,32)(4,20,8,29)(9,22,30,13)(10,18,31,27)(12,25,21,16)(15,28,24,19), (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), (1,3)(5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([[(1,13,5,22),(2,10,6,31),(3,19,7,28),(4,16,8,25),(9,17,30,26),(11,24,32,15),(12,20,21,29),(14,27,23,18)], [(1,17,5,26),(2,14,6,23),(3,11,7,32),(4,20,8,29),(9,22,30,13),(10,18,31,27),(12,25,21,16),(15,28,24,19)], [(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)], [(1,3),(5,7),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21)]])
Matrix representation of Q8:D12 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 41 | 21 |
| 0 | 0 | 52 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 21 |
| 0 | 0 | 40 | 20 |
| 72 | 2 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 72 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,41,52,0,0,21,32],[1,0,0,0,0,1,0,0,0,0,53,40,0,0,21,20],[72,72,0,0,2,1,0,0,0,0,0,72,0,0,1,72],[72,72,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
Q8:D12 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{12} % in TeX
G:=Group("Q8:D12"); // GroupNames label
G:=SmallGroup(192,952);
// by ID
G=gap.SmallGroup(192,952);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b,c*b*c^-1=a,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export