Aliases: C4.3S5, SL2(𝔽5)⋊1C22, C4.A5⋊3C2, C2.7(C2×S5), 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, C2×C4, D4, Q8, C23, D5, C10, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, SL2(𝔽3), C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C8⋊C22, C5⋊C8, C4×D5, GL2(𝔽3), C4.A4, S3×D4, C4.F5, C4.3S4, SL2(𝔽5), C2.S5, C4.A5, C4.3S5
Quotients: C1, C2, C22, S5, C2×S5, 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 C2×S5 |
| 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 C2×S5 |
| 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 C2×S5 |
| 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 C2×S5 |
| ρ15 | 6 | 6 | 0 | 0 | 2 | 0 | -6 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | -1 | -1 | orthogonal lifted from C2×S5 |
| 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