Aliases: Q8⋊D12, C4⋊GL2(𝔽3), SL2(𝔽3)⋊1D4, (C4×Q8)⋊2S3, C2.4(C4⋊S4), (C2×C4).11S4, (C2×Q8).9D6, C22.36(C2×S4), C2.4(C4.3S4), (C4×SL2(𝔽3))⋊6C2, (C2×GL2(𝔽3))⋊4C2, C2.4(C2×GL2(𝔽3)), (C2×SL2(𝔽3)).9C22, SmallGroup(192,952)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — Q8⋊D12 |
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — Q8⋊D12 |
| SL2(𝔽3) — C2×SL2(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, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, SL2(𝔽3), D12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, GL2(𝔽3), C2×SL2(𝔽3), C2×D12, C4⋊SD16, C4×SL2(𝔽3), C2×GL2(𝔽3), Q8⋊D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, GL2(𝔽3), C2×S4, C4⋊S4, C2×GL2(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 C2×S4 |
| 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 C2×S4 |
| 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
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