Aliases: GL2(F3):3C22, CSU2(F3):3C22, SL2(F3).5C23, C4.25(C2xS4), (C2xC4).19S4, C4.6S4:5C2, C4.S4:6C2, C4.3S4:6C2, C4oD4.16D6, C4.A4:5C22, C22.8(C2xS4), (C2xQ8).24D6, C2.16(C22xS4), Q8.D6:3C2, Q8.6(C22xS3), (C2xSL2(F3)):7C22, (C2xC4oD4):4S3, (C2xC4.A4):5C2, SmallGroup(192,1482)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(F3) — GL2(F3):C22 |
Generators and relations for GL2(F3):C22
G = < a,b,c,d,e,f | a4=c3=d2=e2=f2=1, b2=a2, bab-1=eae=dbd=a-1, cac-1=ab, dad=ebe=a2b, af=fa, cbc-1=a, bf=fb, dcd=c-1, ece=ac, cf=fc, ede=fdf=a2d, ef=fe >
Subgroups: 523 in 144 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, Dic3, C12, D6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, SL2(F3), Dic6, C4xS3, D12, C3:D4, C2xC12, C2xM4(2), C4oD8, C8:C22, C8.C22, C2xC4oD4, C2xC4oD4, CSU2(F3), GL2(F3), C2xSL2(F3), C4.A4, C4oD12, D8:C22, Q8.D6, C4.S4, C4.6S4, C4.3S4, C2xC4.A4, GL2(F3):C22
Quotients: C1, C2, C22, S3, C23, D6, S4, C22xS3, C2xS4, C22xS4, GL2(F3):C22
Character table of GL2(F3):C22
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 8 | 1 | 1 | 2 | 6 | 6 | 12 | 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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 3 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | 3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ14 | 3 | 3 | -3 | 1 | -1 | -1 | -1 | 0 | -3 | -3 | 3 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | 3 | 1 | 1 | -1 | 1 | 0 | -3 | -3 | -3 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ16 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | -3 | 1 | -1 | 1 | 1 | 0 | -3 | -3 | 3 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ18 | 3 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 3 | 3 | -3 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ19 | 3 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | 3 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | 3 | 1 | 1 | 1 | -1 | 0 | -3 | -3 | -3 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -i | i | √3 | -√3 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -i | i | -√3 | √3 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | i | -i | √3 | -√3 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | i | -i | -√3 | √3 | complex faithful |
►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 11 3 9)(2 10 4 12)(5 31 7 29)(6 30 8 32)(13 18 15 20)(14 17 16 19)(21 27 23 25)(22 26 24 28)
(2 11 10)(4 9 12)(5 8 30)(6 32 7)(13 19 18)(15 17 20)(22 27 26)(24 25 28)
(1 31)(2 5)(3 29)(4 7)(6 12)(8 10)(9 32)(11 30)(13 25)(14 23)(15 27)(16 21)(17 22)(18 28)(19 24)(20 26)
(1 18)(2 17)(3 20)(4 19)(5 24)(6 23)(7 22)(8 21)(9 15)(10 14)(11 13)(12 16)(25 32)(26 31)(27 30)(28 29)
(5 7)(6 8)(21 23)(22 24)(25 27)(26 28)(29 31)(30 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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,19,18)(15,17,20)(22,27,26)(24,25,28), (1,31)(2,5)(3,29)(4,7)(6,12)(8,10)(9,32)(11,30)(13,25)(14,23)(15,27)(16,21)(17,22)(18,28)(19,24)(20,26), (1,18)(2,17)(3,20)(4,19)(5,24)(6,23)(7,22)(8,21)(9,15)(10,14)(11,13)(12,16)(25,32)(26,31)(27,30)(28,29), (5,7)(6,8)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)>;
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), (1,11,3,9)(2,10,4,12)(5,31,7,29)(6,30,8,32)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28), (2,11,10)(4,9,12)(5,8,30)(6,32,7)(13,19,18)(15,17,20)(22,27,26)(24,25,28), (1,31)(2,5)(3,29)(4,7)(6,12)(8,10)(9,32)(11,30)(13,25)(14,23)(15,27)(16,21)(17,22)(18,28)(19,24)(20,26), (1,18)(2,17)(3,20)(4,19)(5,24)(6,23)(7,22)(8,21)(9,15)(10,14)(11,13)(12,16)(25,32)(26,31)(27,30)(28,29), (5,7)(6,8)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32) );
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)], [(1,11,3,9),(2,10,4,12),(5,31,7,29),(6,30,8,32),(13,18,15,20),(14,17,16,19),(21,27,23,25),(22,26,24,28)], [(2,11,10),(4,9,12),(5,8,30),(6,32,7),(13,19,18),(15,17,20),(22,27,26),(24,25,28)], [(1,31),(2,5),(3,29),(4,7),(6,12),(8,10),(9,32),(11,30),(13,25),(14,23),(15,27),(16,21),(17,22),(18,28),(19,24),(20,26)], [(1,18),(2,17),(3,20),(4,19),(5,24),(6,23),(7,22),(8,21),(9,15),(10,14),(11,13),(12,16),(25,32),(26,31),(27,30),(28,29)], [(5,7),(6,8),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)]])
Matrix representation of GL2(F3):C22 ►in GL4(F5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 2 |
| 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 3 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 3 | 0 | 4 |
| 3 | 0 | 2 | 0 |
| 0 | 2 | 0 | 4 |
| 1 | 0 | 1 | 0 |
| 0 | 2 | 0 | 2 |
| 0 | 2 | 0 | 4 |
| 3 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 2 | 0 | 2 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 3 | 0 |
| 0 | 2 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [3,0,4,0,0,2,0,0,0,0,2,0,0,2,0,3],[3,0,0,0,0,1,0,3,3,0,2,0,0,1,0,4],[3,0,1,0,0,2,0,2,2,0,1,0,0,4,0,2],[0,3,0,0,2,0,0,0,0,2,0,4,4,0,4,0],[2,0,1,0,0,4,0,2,2,0,3,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1] >;
GL2(F3):C22 in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_3)\rtimes C_2^2 % in TeX
G:=Group("GL(2,3):C2^2"); // GroupNames label
G:=SmallGroup(192,1482);
// by ID
G=gap.SmallGroup(192,1482);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^3=d^2=e^2=f^2=1,b^2=a^2,b*a*b^-1=e*a*e=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=e*b*e=a^2*b,a*f=f*a,c*b*c^-1=a,b*f=f*b,d*c*d=c^-1,e*c*e=a*c,c*f=f*c,e*d*e=f*d*f=a^2*d,e*f=f*e>;
// generators/relations
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