Aliases: Q8.2D12, SL2(𝔽3).2D4, (C2×C4).4S4, (C4×Q8)⋊4S3, C2.5(C4⋊S4), (C2×Q8).11D6, C22.37(C2×S4), C2.5(C4.6S4), (C4×SL2(𝔽3))⋊2C2, (C2×CSU2(𝔽3))⋊1C2, C2.5(Q8.D6), (C2×GL2(𝔽3)).2C2, (C2×SL2(𝔽3)).11C22, SmallGroup(192,954)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — Q8.2D12 |
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — Q8.2D12 |
| SL2(𝔽3) — C2×SL2(𝔽3) — Q8.2D12 |
Generators and relations for Q8.2D12
G = < a,b,c,d | a4=c12=1, b2=d2=a2, bab-1=dad-1=a-1, cac-1=ab, cbc-1=a, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 329 in 73 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, D6⋊C4, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), Q8.D4, C4×SL2(𝔽3), C2×CSU2(𝔽3), C2×GL2(𝔽3), Q8.2D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2×S4, C4⋊S4, Q8.D6, C4.6S4, Q8.2D12
Character table of Q8.2D12
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 24 | 8 | 2 | 2 | 6 | 6 | 12 | 24 | 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 | -1 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | -√-2 | √2 | √-2 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √2 | √-2 | -√2 | -√-2 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √2 | -√-2 | -√2 | √-2 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | √-2 | √2 | -√-2 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ14 | 3 | 3 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 3 | 3 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | 0 | -3 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 3 | 3 | 1 | 0 | -3 | -3 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from Q8.D6 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from Q8.D6 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ23 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
►On 32 points
(1 24 8 19)(2 21 5 16)(3 30 6 13)(4 27 7 10)(9 17 26 22)(11 32 28 15)(12 20 29 25)(14 23 31 18)
(1 28 8 11)(2 25 5 20)(3 22 6 17)(4 31 7 14)(9 30 26 13)(10 18 27 23)(12 21 29 16)(15 24 32 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 2 8 5)(3 4 6 7)(9 31 26 14)(10 13 27 30)(11 29 28 12)(15 25 32 20)(16 19 21 24)(17 23 22 18)
G:=sub<Sym(32)| (1,24,8,19)(2,21,5,16)(3,30,6,13)(4,27,7,10)(9,17,26,22)(11,32,28,15)(12,20,29,25)(14,23,31,18), (1,28,8,11)(2,25,5,20)(3,22,6,17)(4,31,7,14)(9,30,26,13)(10,18,27,23)(12,21,29,16)(15,24,32,19), (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), (1,2,8,5)(3,4,6,7)(9,31,26,14)(10,13,27,30)(11,29,28,12)(15,25,32,20)(16,19,21,24)(17,23,22,18)>;
G:=Group( (1,24,8,19)(2,21,5,16)(3,30,6,13)(4,27,7,10)(9,17,26,22)(11,32,28,15)(12,20,29,25)(14,23,31,18), (1,28,8,11)(2,25,5,20)(3,22,6,17)(4,31,7,14)(9,30,26,13)(10,18,27,23)(12,21,29,16)(15,24,32,19), (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), (1,2,8,5)(3,4,6,7)(9,31,26,14)(10,13,27,30)(11,29,28,12)(15,25,32,20)(16,19,21,24)(17,23,22,18) );
G=PermutationGroup([[(1,24,8,19),(2,21,5,16),(3,30,6,13),(4,27,7,10),(9,17,26,22),(11,32,28,15),(12,20,29,25),(14,23,31,18)], [(1,28,8,11),(2,25,5,20),(3,22,6,17),(4,31,7,14),(9,30,26,13),(10,18,27,23),(12,21,29,16),(15,24,32,19)], [(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)], [(1,2,8,5),(3,4,6,7),(9,31,26,14),(10,13,27,30),(11,29,28,12),(15,25,32,20),(16,19,21,24),(17,23,22,18)]])
Matrix representation of Q8.2D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 29 |
| 0 | 0 | 56 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 29 |
| 0 | 0 | 44 | 16 |
| 28 | 69 | 0 | 0 |
| 32 | 45 | 0 | 0 |
| 0 | 0 | 47 | 67 |
| 0 | 0 | 47 | 53 |
| 28 | 69 | 0 | 0 |
| 68 | 45 | 0 | 0 |
| 0 | 0 | 47 | 67 |
| 0 | 0 | 52 | 26 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,45,56,0,0,29,28],[1,0,0,0,0,1,0,0,0,0,57,44,0,0,29,16],[28,32,0,0,69,45,0,0,0,0,47,47,0,0,67,53],[28,68,0,0,69,45,0,0,0,0,47,52,0,0,67,26] >;
Q8.2D12 in GAP, Magma, Sage, TeX
Q_8._2D_{12} % in TeX
G:=Group("Q8.2D12"); // GroupNames label
G:=SmallGroup(192,954);
// by ID
G=gap.SmallGroup(192,954);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,708,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=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
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