direct product, non-abelian, not soluble
Aliases: C2×SL2(𝔽5), C22.A5, C2.3(C2×A5), SmallGroup(240,94)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C2×SL2(𝔽5) |
| SL2(𝔽5) — C2×SL2(𝔽5) |
10C3
6C5
15C4
15C4
10C6
10C6
10C6
6C10
6C10
6C10
5Q8
15Q8
15C2×C4
10Dic3
10C2×C6
10Dic3
6Dic5
6Dic5
10C2×Dic3
Character table of C2×SL2(𝔽5)
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 1 | 1 | 20 | 30 | 30 | 12 | 12 | 20 | 20 | 20 | 12 | 12 | 12 | 12 | 12 | 12 | |
| 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 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1 | -1 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ4 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | -1 | 1 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ5 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | -1 | 1 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ6 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1 | -1 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ7 | 3 | -3 | 3 | -3 | 0 | -1 | 1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2×A5 |
| ρ8 | 3 | -3 | 3 | -3 | 0 | -1 | 1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2×A5 |
| ρ9 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ10 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ11 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A5 |
| ρ12 | 4 | 4 | 4 | 4 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ13 | 4 | 4 | -4 | -4 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ14 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ15 | 5 | -5 | 5 | -5 | -1 | 1 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ16 | 5 | 5 | 5 | 5 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ17 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | symplectic lifted from SL2(𝔽5), Schur index 2 |
| ρ18 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | symplectic lifted from SL2(𝔽5), 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 26 40 47 25 3 12 30 37 11)(2 21 45 42 20 4 17 35 32 16)(5 29 22 19 38 7 39 18 15 48)(6 34 27 24 33 8 44 13 10 43)(9 23)(14 28)(31 41)(36 46)
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), (1,26,40,47,25,3,12,30,37,11)(2,21,45,42,20,4,17,35,32,16)(5,29,22,19,38,7,39,18,15,48)(6,34,27,24,33,8,44,13,10,43)(9,23)(14,28)(31,41)(36,46)>;
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), (1,26,40,47,25,3,12,30,37,11)(2,21,45,42,20,4,17,35,32,16)(5,29,22,19,38,7,39,18,15,48)(6,34,27,24,33,8,44,13,10,43)(9,23)(14,28)(31,41)(36,46) );
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)], [(1,26,40,47,25,3,12,30,37,11),(2,21,45,42,20,4,17,35,32,16),(5,29,22,19,38,7,39,18,15,48),(6,34,27,24,33,8,44,13,10,43),(9,23),(14,28),(31,41),(36,46)]])
C2×SL2(𝔽5) is a maximal subgroup of
C22.2S5 C22.S5 D4.A5
Matrix representation of C2×SL2(𝔽5) ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 7 | 22 |
| 0 | 53 | 36 |
| 60 | 0 | 0 |
| 0 | 20 | 30 |
| 0 | 18 | 24 |
G:=sub<GL(3,GF(61))| [60,0,0,0,7,53,0,22,36],[60,0,0,0,20,18,0,30,24] >;
C2×SL2(𝔽5) in GAP, Magma, Sage, TeX
C_2\times {\rm SL}_2({\mathbb F}_5) % in TeX
G:=Group("C2xSL(2,5)"); // GroupNames label
G:=SmallGroup(240,94);
// by ID
G=gap.SmallGroup(240,94);
# by ID
Export
Subgroup lattice of C2×SL2(𝔽5) in TeX
Character table of C2×SL2(𝔽5) in TeX