Aliases: C4.3S5, SL2(F5):1C22, C4.A5:3C2, C2.7(C2xS5), C2.S5:1C2, SmallGroup(480,948)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(F5) — C4.3S5 |
| SL2(F5) — C4.3S5 |
Subgroups: 934 in 76 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2xC4, D4, Q8, C23, D5, C10, Dic3, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, Dic5, C20, D10, SL2(F3), C4xS3, D12, C3:D4, C3xD4, C22xS3, C8:C22, C5:C8, C4xD5, GL2(F3), C4.A4, S3xD4, C4.F5, C4.3S4, SL2(F5), C2.S5, C4.A5, C4.3S5
Quotients: C1, C2, C22, S5, C2xS5, C4.3S5
Character table of C4.3S5
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5 | 6A | 6B | 6C | 8A | 8B | 10 | 12 | 20A | 20B | |
| size | 1 | 1 | 20 | 20 | 30 | 20 | 2 | 30 | 24 | 20 | 40 | 40 | 60 | 60 | 24 | 40 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 4 | 4 | 2 | 2 | 0 | 1 | 4 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | -1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ6 | 4 | 4 | 2 | -2 | 0 | 1 | -4 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xS5 |
| ρ7 | 4 | 4 | -2 | -2 | 0 | 1 | 4 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | 4 | -2 | 2 | 0 | 1 | -4 | 0 | -1 | 1 | 1 | -1 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xS5 |
| ρ9 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | √-5 | -√-5 | complex faithful |
| ρ10 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | -√-5 | √-5 | complex faithful |
| ρ11 | 5 | 5 | -1 | 1 | -1 | -1 | -5 | 1 | 0 | -1 | -1 | 1 | 1 | -1 | 0 | 1 | 0 | 0 | orthogonal lifted from C2xS5 |
| ρ12 | 5 | 5 | 1 | 1 | 1 | -1 | 5 | 1 | 0 | -1 | 1 | 1 | -1 | -1 | 0 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ13 | 5 | 5 | -1 | -1 | 1 | -1 | 5 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ14 | 5 | 5 | 1 | -1 | -1 | -1 | -5 | 1 | 0 | -1 | 1 | -1 | -1 | 1 | 0 | 1 | 0 | 0 | orthogonal lifted from C2xS5 |
| ρ15 | 6 | 6 | 0 | 0 | 2 | 0 | -6 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | -1 | -1 | orthogonal lifted from C2xS5 |
| ρ16 | 6 | 6 | 0 | 0 | -2 | 0 | 6 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ17 | 8 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal faithful |
| ρ18 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal faithful |
►On 40 points
Generators in S40
(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)(33 34 35 36 37 38 39 40)
(1 3 5 7)(2 34 6 38)(4 21 8 17)(9 33 13 37)(10 22 14 18)(11 32 15 28)(12 27 16 31)(19 36 23 40)(20 25 24 29)(26 39 30 35)
(1 5)(3 7)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)
G:=sub<Sym(40)| (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)(33,34,35,36,37,38,39,40), (1,3,5,7)(2,34,6,38)(4,21,8,17)(9,33,13,37)(10,22,14,18)(11,32,15,28)(12,27,16,31)(19,36,23,40)(20,25,24,29)(26,39,30,35), (1,5)(3,7)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)>;
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)(33,34,35,36,37,38,39,40), (1,3,5,7)(2,34,6,38)(4,21,8,17)(9,33,13,37)(10,22,14,18)(11,32,15,28)(12,27,16,31)(19,36,23,40)(20,25,24,29)(26,39,30,35), (1,5)(3,7)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35) );
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),(33,34,35,36,37,38,39,40)], [(1,3,5,7),(2,34,6,38),(4,21,8,17),(9,33,13,37),(10,22,14,18),(11,32,15,28),(12,27,16,31),(19,36,23,40),(20,25,24,29),(26,39,30,35)], [(1,5),(3,7),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35)]])
Matrix representation of C4.3S5 ►in GL4(F3) generated by
| 2 | 1 | 0 | 1 |
| 0 | 2 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 2 | 0 | 2 |
| 2 | 2 | 1 | 1 |
| 1 | 0 | 2 | 0 |
| 2 | 0 | 1 | 1 |
| 0 | 2 | 2 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 |
| 2 | 0 | 1 | 2 |
| 1 | 0 | 0 | 2 |
G:=sub<GL(4,GF(3))| [2,0,0,2,1,2,1,2,0,1,0,0,1,0,0,2],[2,1,2,0,2,0,0,2,1,2,1,2,1,0,1,0],[1,1,2,1,0,2,0,0,0,0,1,0,0,0,2,2] >;
C4.3S5 in GAP, Magma, Sage, TeX
C_4._3S_5
% in TeX
G:=Group("C4.3S5"); // GroupNames label
G:=SmallGroup(480,948);
// by ID
G=gap.SmallGroup(480,948);
# by ID
Export