Aliases: C22.S5, CSU2(𝔽5)⋊2C2, SL2(𝔽5).4C22, C2.11(C2×S5), C2.S5⋊2C2, (C2×SL2(𝔽5))⋊3C2, SmallGroup(480,953)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — C2 — C22 — C2×SL2(𝔽5) — C22.S5 |
| SL2(𝔽5) — C22.S5 |
| SL2(𝔽5) — C22.S5 |
Subgroups: 646 in 66 conjugacy classes, 8 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C10, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C2×C10, SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C8.C22, C5⋊C8, C2×Dic5, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), D4⋊2S3, C22.F5, Q8.D6, SL2(𝔽5), CSU2(𝔽5), C2.S5, C2×SL2(𝔽5), C22.S5
Quotients: C1, C2, C22, S5, C2×S5, C22.S5
Character table of C22.S5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 12 | |
| size | 1 | 1 | 2 | 20 | 20 | 20 | 30 | 30 | 24 | 20 | 40 | 40 | 60 | 60 | 24 | 24 | 24 | 40 | |
| 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 | 4 | -2 | 1 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 1 | orthogonal lifted from S5 |
| ρ6 | 4 | 4 | 4 | 2 | 1 | 2 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ7 | 4 | 4 | -4 | 2 | 1 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 1 | orthogonal lifted from C2×S5 |
| ρ8 | 4 | 4 | -4 | -2 | 1 | 2 | 0 | 0 | -1 | 1 | 1 | -1 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ9 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 0 | symplectic faithful, Schur index 2 |
| ρ10 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 0 | symplectic faithful, Schur index 2 |
| ρ11 | 5 | 5 | -5 | 1 | -1 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S5 |
| ρ12 | 5 | 5 | -5 | -1 | -1 | 1 | 1 | -1 | 0 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×S5 |
| ρ13 | 5 | 5 | 5 | 1 | -1 | 1 | 1 | 1 | 0 | -1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | orthogonal lifted from S5 |
| ρ14 | 5 | 5 | 5 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | orthogonal lifted from S5 |
| ρ15 | 6 | 6 | 6 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | orthogonal lifted from S5 |
| ρ16 | 6 | 6 | -6 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | orthogonal lifted from C2×S5 |
| ρ17 | 8 | -8 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | symplectic faithful, Schur index 2 |
| ρ18 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(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)(41 42 43 44 45 46 47 48)
(1 41 31 6 38 37)(2 34 33 5 45 27)(3 9 48 28 10 8)(4 7 13 44 32 14)(11 24 43 29 21 12)(15 20 47 25 17 16)(18 39 42 26 40 23)(19 22 35 46 30 36)
G:=sub<Sym(48)| (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)(41,42,43,44,45,46,47,48), (1,41,31,6,38,37)(2,34,33,5,45,27)(3,9,48,28,10,8)(4,7,13,44,32,14)(11,24,43,29,21,12)(15,20,47,25,17,16)(18,39,42,26,40,23)(19,22,35,46,30,36)>;
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)(41,42,43,44,45,46,47,48), (1,41,31,6,38,37)(2,34,33,5,45,27)(3,9,48,28,10,8)(4,7,13,44,32,14)(11,24,43,29,21,12)(15,20,47,25,17,16)(18,39,42,26,40,23)(19,22,35,46,30,36) );
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),(41,42,43,44,45,46,47,48)], [(1,41,31,6,38,37),(2,34,33,5,45,27),(3,9,48,28,10,8),(4,7,13,44,32,14),(11,24,43,29,21,12),(15,20,47,25,17,16),(18,39,42,26,40,23),(19,22,35,46,30,36)]])
Matrix representation of C22.S5 ►in GL4(𝔽5) generated by
| 2 | 2 | 3 | 1 |
| 1 | 0 | 1 | 2 |
| 2 | 1 | 4 | 0 |
| 3 | 0 | 3 | 4 |
| 3 | 4 | 0 | 0 |
| 0 | 3 | 3 | 3 |
| 4 | 0 | 1 | 4 |
| 0 | 1 | 3 | 3 |
G:=sub<GL(4,GF(5))| [2,1,2,3,2,0,1,0,3,1,4,3,1,2,0,4],[3,0,4,0,4,3,0,1,0,3,1,3,0,3,4,3] >;
C22.S5 in GAP, Magma, Sage, TeX
C_2^2.S_5
% in TeX
G:=Group("C2^2.S5"); // GroupNames label
G:=SmallGroup(480,953);
// by ID
G=gap.SmallGroup(480,953);
# by ID
Export