Aliases: Q8.7S4, 2+ 1+4:3S3, GL2(F3):4C22, CSU2(F3):7C22, SL2(F3).7C23, C4.13(C2xS4), C4oD4.6D6, Q8.A4:3C2, C4.3S4:4C2, C4.6S4:2C2, C4.A4:1C22, C2.18(C22xS4), Q8.8(C22xS3), SmallGroup(192,1484)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(F3) — Q8.7S4 |
Generators and relations for Q8.7S4
G = < a,b,c,d,e,f | a4=e3=f2=1, b2=c2=d2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, 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: 599 in 146 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, Q8, Q8, C23, Dic3, C12, D6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C4oD4, C4oD4, SL2(F3), C4xS3, D12, C3xQ8, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, 2+ 1+4, CSU2(F3), GL2(F3), C4.A4, Q8:3S3, D4oD8, C4.6S4, C4.3S4, Q8.A4, Q8.7S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22xS3, C2xS4, C22xS4, Q8.7S4
Character table of Q8.7S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | |
| size | 1 | 1 | 6 | 6 | 6 | 12 | 12 | 12 | 8 | 2 | 2 | 2 | 6 | 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 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 3 | 3 | 1 | -1 | 1 | 1 | -1 | 1 | 0 | -3 | -3 | 3 | -1 | -1 | 0 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ14 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | 1 | 0 | 3 | 3 | 3 | -1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | -3 | -3 | 3 | -1 | 1 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ16 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 1 | 1 | -1 | 1 | 1 | -1 | 0 | 3 | -3 | -3 | -1 | -1 | 0 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ18 | 3 | 3 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | -3 | 3 | -3 | -1 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ19 | 3 | 3 | 1 | 1 | -1 | -1 | -1 | 1 | 0 | 3 | -3 | -3 | -1 | 1 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ20 | 3 | 3 | -1 | 1 | 1 | -1 | 1 | 1 | 0 | -3 | 3 | -3 | -1 | -1 | 0 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal 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)
(2 4)(5 16)(6 15)(7 14)(8 13)(9 11)(18 20)(21 23)(25 32)(26 31)(27 30)(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), (2,4)(5,16)(6,15)(7,14)(8,13)(9,11)(18,20)(21,23)(25,32)(26,31)(27,30)(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), (2,4)(5,16)(6,15)(7,14)(8,13)(9,11)(18,20)(21,23)(25,32)(26,31)(27,30)(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)], [(2,4),(5,16),(6,15),(7,14),(8,13),(9,11),(18,20),(21,23),(25,32),(26,31),(27,30),(28,29)]])
Matrix representation of Q8.7S4 ►in GL4(F7) generated by
| 3 | 1 | 4 | 3 |
| 5 | 3 | 0 | 5 |
| 6 | 6 | 0 | 1 |
| 1 | 4 | 3 | 1 |
| 6 | 3 | 6 | 4 |
| 5 | 3 | 1 | 6 |
| 2 | 2 | 6 | 3 |
| 5 | 6 | 4 | 6 |
| 5 | 1 | 3 | 2 |
| 3 | 5 | 6 | 6 |
| 2 | 2 | 6 | 3 |
| 0 | 6 | 5 | 5 |
| 0 | 2 | 1 | 5 |
| 4 | 4 | 6 | 5 |
| 0 | 5 | 1 | 1 |
| 1 | 3 | 3 | 2 |
| 1 | 2 | 2 | 6 |
| 1 | 0 | 4 | 4 |
| 4 | 6 | 6 | 3 |
| 6 | 2 | 3 | 5 |
| 0 | 2 | 3 | 0 |
| 4 | 6 | 0 | 4 |
| 0 | 3 | 0 | 2 |
| 1 | 5 | 4 | 1 |
G:=sub<GL(4,GF(7))| [3,5,6,1,1,3,6,4,4,0,0,3,3,5,1,1],[6,5,2,5,3,3,2,6,6,1,6,4,4,6,3,6],[5,3,2,0,1,5,2,6,3,6,6,5,2,6,3,5],[0,4,0,1,2,4,5,3,1,6,1,3,5,5,1,2],[1,1,4,6,2,0,6,2,2,4,6,3,6,4,3,5],[0,4,0,1,2,6,3,5,3,0,0,4,0,4,2,1] >;
Q8.7S4 in GAP, Magma, Sage, TeX
Q_8._7S_4
% in TeX
G:=Group("Q8.7S4"); // GroupNames label
G:=SmallGroup(192,1484);
// by ID
G=gap.SmallGroup(192,1484);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,1059,520,451,1684,655,172,1013,404,285,124]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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