Aliases: C2.3S5, SL2(𝔽5)⋊C2, SmallGroup(240,90)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C2.S5 |
| SL2(𝔽5) — C2.S5 |
Character table of C2.S5
| class | 1 | 2A | 2B | 3 | 4 | 5 | 6A | 6B | 6C | 8A | 8B | 10 | |
| size | 1 | 1 | 20 | 20 | 30 | 24 | 20 | 20 | 20 | 30 | 30 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 4 | 4 | -2 | 1 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | -1 | orthogonal lifted from S5 |
| ρ4 | 4 | 4 | 2 | 1 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | -1 | orthogonal lifted from S5 |
| ρ5 | 4 | -4 | 0 | -2 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | 1 | symplectic faithful, Schur index 2 |
| ρ6 | 4 | -4 | 0 | 1 | 0 | -1 | √-3 | -√-3 | -1 | 0 | 0 | 1 | complex faithful |
| ρ7 | 4 | -4 | 0 | 1 | 0 | -1 | -√-3 | √-3 | -1 | 0 | 0 | 1 | complex faithful |
| ρ8 | 5 | 5 | 1 | -1 | 1 | 0 | 1 | 1 | -1 | -1 | -1 | 0 | orthogonal lifted from S5 |
| ρ9 | 5 | 5 | -1 | -1 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from S5 |
| ρ10 | 6 | 6 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S5 |
| ρ11 | 6 | -6 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -√-2 | √-2 | -1 | complex faithful |
| ρ12 | 6 | -6 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | √-2 | -√-2 | -1 | complex 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 32 6 28)(4 20 8 24)(9 21 13 17)(10 15 14 11)(12 19 16 23)(18 26 22 30)(25 35 29 39)(27 38 31 34)(33 36 37 40)
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)(2,32,6,28)(4,20,8,24)(9,21,13,17)(10,15,14,11)(12,19,16,23)(18,26,22,30)(25,35,29,39)(27,38,31,34)(33,36,37,40)>;
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)(2,32,6,28)(4,20,8,24)(9,21,13,17)(10,15,14,11)(12,19,16,23)(18,26,22,30)(25,35,29,39)(27,38,31,34)(33,36,37,40) );
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),(2,32,6,28),(4,20,8,24),(9,21,13,17),(10,15,14,11),(12,19,16,23),(18,26,22,30),(25,35,29,39),(27,38,31,34),(33,36,37,40)]])
C2.S5 is a maximal subgroup of
C4.6S5 C4.3S5 C22.S5
C2.S5 is a maximal quotient of C22.2S5
Matrix representation of C2.S5 ►in GL4(𝔽3) generated by
| 1 | 0 | 2 | 1 |
| 0 | 1 | 2 | 0 |
| 1 | 1 | 0 | 1 |
| 2 | 1 | 0 | 1 |
| 2 | 2 | 1 | 0 |
| 2 | 0 | 0 | 2 |
| 0 | 2 | 1 | 2 |
| 1 | 2 | 2 | 0 |
G:=sub<GL(4,GF(3))| [1,0,1,2,0,1,1,1,2,2,0,0,1,0,1,1],[2,2,0,1,2,0,2,2,1,0,1,2,0,2,2,0] >;
C2.S5 in GAP, Magma, Sage, TeX
C_2.S_5
% in TeX
G:=Group("C2.S5"); // GroupNames label
G:=SmallGroup(240,90);
// by ID
G=gap.SmallGroup(240,90);
# by ID
Export
Subgroup lattice of C2.S5 in TeX
Character table of C2.S5 in TeX