Aliases: Q8.6S4, 2+ 1+4⋊2S3, GL2(𝔽3)⋊7C22, CSU2(𝔽3)⋊4C22, SL2(𝔽3).6C23, C4.12(C2×S4), C4○D4.5D6, Q8.A4⋊2C2, C4.S4⋊4C2, C4.6S4⋊1C2, C2.17(C22×S4), C4.A4.5C22, Q8.7(C22×S3), SmallGroup(192,1483)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — Q8.6S4 |
Generators and relations for Q8.6S4
G = < a,b,c,d,e,f | a4=e3=f2=1, b2=c2=d2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >
Subgroups: 487 in 140 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), Dic6, C4×S3, C3×Q8, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, CSU2(𝔽3), GL2(𝔽3), C4.A4, S3×Q8, D4○SD16, C4.S4, C4.6S4, Q8.A4, Q8.6S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C22×S4, Q8.6S4
Character table of Q8.6S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | |
| size | 1 | 1 | 6 | 6 | 6 | 12 | 8 | 2 | 2 | 2 | 6 | 12 | 12 | 12 | 8 | 6 | 6 | 12 | 12 | 12 | 16 | 16 | 16 | |
| 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 3 | 3 | 1 | 1 | -1 | -1 | 0 | -3 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | -1 | 1 | 1 | 1 | 0 | 3 | -3 | -3 | -1 | 1 | -1 | -1 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 1 | -1 | 1 | 1 | 0 | -3 | -3 | 3 | -1 | -1 | -1 | 1 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 1 | 1 | -1 | 1 | 0 | -3 | 3 | -3 | -1 | -1 | 1 | -1 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | -1 | 1 | 1 | -1 | 0 | 3 | -3 | -3 | -1 | -1 | 1 | 1 | 0 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ19 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | 3 | 3 | 3 | -1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | 1 | -1 | 1 | -1 | 0 | -3 | -3 | 3 | -1 | 1 | 1 | -1 | 0 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
Generators in S32
(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 24 3 22)(2 23 4 21)(5 27 7 25)(6 26 8 28)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)
(1 8 3 6)(2 5 4 7)(9 31 11 29)(10 32 12 30)(13 18 15 20)(14 19 16 17)(21 25 23 27)(22 26 24 28)
(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 22 11 24)(10 23 12 21)(25 30 27 32)(26 31 28 29)
(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 31 28)(10 32 25)(11 29 26)(12 30 27)
(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 19)(18 20)(25 30)(26 31)(27 32)(28 29)
G:=sub<Sym(32)| (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), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(25,30)(26,31)(27,32)(28,29)>;
G:=Group( (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), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,19)(18,20)(25,30)(26,31)(27,32)(28,29) );
G=PermutationGroup([[(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)], [(1,24,3,22),(2,23,4,21),(5,27,7,25),(6,26,8,28),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)], [(1,8,3,6),(2,5,4,7),(9,31,11,29),(10,32,12,30),(13,18,15,20),(14,19,16,17),(21,25,23,27),(22,26,24,28)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,22,11,24),(10,23,12,21),(25,30,27,32),(26,31,28,29)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,31,28),(10,32,25),(11,29,26),(12,30,27)], [(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,19),(18,20),(25,30),(26,31),(27,32),(28,29)]])
Matrix representation of Q8.6S4 ►in GL4(𝔽3) generated by
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 2 | 1 |
| 1 | 2 | 1 | 1 |
| 2 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 2 |
| 1 | 1 | 1 | 0 |
| 2 | 0 | 1 | 0 |
| 0 | 0 | 1 | 2 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 |
| 2 | 0 | 2 | 0 |
| 0 | 1 | 2 | 1 |
| 2 | 0 | 1 | 0 |
| 2 | 1 | 2 | 2 |
| 2 | 0 | 2 | 0 |
| 0 | 1 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 2 | 1 |
| 1 | 0 | 1 | 0 |
| 0 | 2 | 1 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(3))| [0,1,0,1,2,0,2,2,0,0,2,1,0,0,1,1],[2,1,0,1,1,1,1,1,0,0,0,1,0,0,2,0],[2,0,1,1,0,0,0,1,1,1,1,1,0,2,0,0],[2,0,2,2,0,1,0,1,2,2,1,2,0,1,0,2],[2,0,1,1,0,1,0,0,2,2,0,2,0,1,0,1],[1,0,0,0,0,2,0,0,1,1,2,1,0,2,0,1] >;
Q8.6S4 in GAP, Magma, Sage, TeX
Q_8._6S_4
% in TeX
G:=Group("Q8.6S4"); // GroupNames label
G:=SmallGroup(192,1483);
// by ID
G=gap.SmallGroup(192,1483);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,1059,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=e^3=f^2=1,b^2=c^2=d^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^2*d,f*e*f=e^-1>;
// generators/relations
Export