Aliases: D4.5S4, 2- 1+4:3S3, GL2(F3).C22, CSU2(F3).C22, SL2(F3).9C23, (C2xQ8).D6, C4.15(C2xS4), D4.A4:3C2, C4oD4.7D6, C4.6S4:4C2, C4.S4:5C2, C22.6(C2xS4), C2.20(C22xS4), Q8.D6:2C2, C4.A4.6C22, Q8.10(C22xS3), (C2xCSU2(F3)):6C2, (C2xSL2(F3)).C22, SmallGroup(192,1486)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(F3) — D4.5S4 |
Generators and relations for D4.5S4
G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a2b, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >
Subgroups: 471 in 140 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, D4, D4, Q8, Q8, Dic3, C12, D6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C2xQ8, C2xQ8, C4oD4, C4oD4, SL2(F3), Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C8oD4, C2xQ16, C4oD8, C8.C22, 2- 1+4, 2- 1+4, CSU2(F3), CSU2(F3), GL2(F3), C2xSL2(F3), C4.A4, D4:2S3, Q8oD8, C2xCSU2(F3), Q8.D6, C4.S4, C4.6S4, D4.A4, D4.5S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22xS3, C2xS4, C22xS4, D4.5S4
Character table of D4.5S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12 | |
| size | 1 | 1 | 2 | 2 | 6 | 12 | 8 | 2 | 6 | 6 | 6 | 12 | 12 | 12 | 8 | 16 | 16 | 6 | 6 | 12 | 12 | 12 | 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 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 0 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ13 | 3 | 3 | -3 | 3 | 1 | 1 | 0 | -3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 0 | orthogonal lifted from C2xS4 |
| ρ14 | 3 | 3 | -3 | -3 | -1 | 1 | 0 | 3 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | 3 | -3 | 1 | -1 | 0 | -3 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from C2xS4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | -3 | 3 | 1 | -1 | 0 | -3 | -1 | -1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | orthogonal lifted from C2xS4 |
| ρ18 | 3 | 3 | -3 | -3 | -1 | -1 | 0 | 3 | 1 | -1 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from C2xS4 |
| ρ19 | 3 | 3 | 3 | -3 | 1 | 1 | 0 | -3 | 1 | -1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 0 | orthogonal lifted from C2xS4 |
| ρ20 | 3 | 3 | 3 | 3 | -1 | 1 | 0 | 3 | -1 | -1 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | orthogonal lifted from S4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| 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 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 26)(27 28)(29 30)(31 32)
(1 8 3 6)(2 5 4 7)(9 24 11 22)(10 21 12 23)(13 19 15 17)(14 20 16 18)(25 29 27 31)(26 30 28 32)
(1 19 3 17)(2 20 4 18)(5 14 7 16)(6 15 8 13)(9 25 11 27)(10 26 12 28)(21 32 23 30)(22 29 24 31)
(5 20 16)(6 17 13)(7 18 14)(8 19 15)(9 24 31)(10 21 32)(11 22 29)(12 23 30)
(1 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)
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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32), (1,8,3,6)(2,5,4,7)(9,24,11,22)(10,21,12,23)(13,19,15,17)(14,20,16,18)(25,29,27,31)(26,30,28,32), (1,19,3,17)(2,20,4,18)(5,14,7,16)(6,15,8,13)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,20,16)(6,17,13)(7,18,14)(8,19,15)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32)>;
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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32), (1,8,3,6)(2,5,4,7)(9,24,11,22)(10,21,12,23)(13,19,15,17)(14,20,16,18)(25,29,27,31)(26,30,28,32), (1,19,3,17)(2,20,4,18)(5,14,7,16)(6,15,8,13)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,20,16)(6,17,13)(7,18,14)(8,19,15)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32) );
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)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,26),(27,28),(29,30),(31,32)], [(1,8,3,6),(2,5,4,7),(9,24,11,22),(10,21,12,23),(13,19,15,17),(14,20,16,18),(25,29,27,31),(26,30,28,32)], [(1,19,3,17),(2,20,4,18),(5,14,7,16),(6,15,8,13),(9,25,11,27),(10,26,12,28),(21,32,23,30),(22,29,24,31)], [(5,20,16),(6,17,13),(7,18,14),(8,19,15),(9,24,31),(10,21,32),(11,22,29),(12,23,30)], [(1,25),(2,26),(3,27),(4,28),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)]])
Matrix representation of D4.5S4 ►in GL4(F7) generated by
| 6 | 2 | 0 | 5 |
| 5 | 0 | 4 | 3 |
| 2 | 5 | 4 | 3 |
| 6 | 6 | 4 | 4 |
| 4 | 0 | 5 | 2 |
| 5 | 0 | 0 | 3 |
| 2 | 5 | 4 | 3 |
| 5 | 5 | 1 | 6 |
| 6 | 6 | 1 | 1 |
| 2 | 0 | 4 | 1 |
| 3 | 3 | 0 | 1 |
| 4 | 3 | 5 | 1 |
| 4 | 0 | 6 | 5 |
| 6 | 0 | 1 | 1 |
| 5 | 5 | 2 | 4 |
| 6 | 1 | 4 | 1 |
| 1 | 1 | 4 | 1 |
| 0 | 3 | 3 | 3 |
| 6 | 6 | 0 | 2 |
| 1 | 6 | 3 | 1 |
| 6 | 6 | 3 | 6 |
| 6 | 3 | 4 | 4 |
| 6 | 3 | 6 | 3 |
| 5 | 0 | 4 | 6 |
G:=sub<GL(4,GF(7))| [6,5,2,6,2,0,5,6,0,4,4,4,5,3,3,4],[4,5,2,5,0,0,5,5,5,0,4,1,2,3,3,6],[6,2,3,4,6,0,3,3,1,4,0,5,1,1,1,1],[4,6,5,6,0,0,5,1,6,1,2,4,5,1,4,1],[1,0,6,1,1,3,6,6,4,3,0,3,1,3,2,1],[6,6,6,5,6,3,3,0,3,4,6,4,6,4,3,6] >;
D4.5S4 in GAP, Magma, Sage, TeX
D_4._5S_4
% in TeX
G:=Group("D4.5S4"); // GroupNames label
G:=SmallGroup(192,1486);
// by ID
G=gap.SmallGroup(192,1486);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,1059,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=f^2=1,c^2=d^2=a^2,b*a*b=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,f*b*f=a^2*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