Aliases: C4.A5, SL2(𝔽5)⋊2C2, C2.2(C2×A5), SmallGroup(240,93)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C4.A5 |
| SL2(𝔽5) — C4.A5 |
30C2
10C3
6C5
15C22
15C4
10C6
10S3
10S3
6C10
6D5
6D5
5Q8
15D4
15C2×C4
10D6
10Dic3
10C12
6C20
6Dic5
6D10
10C4×S3
Character table of C4.A5
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 30 | 20 | 1 | 1 | 30 | 12 | 12 | 20 | 12 | 12 | 20 | 20 | 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 | 0 | -1 | -2i | 2i | 0 | -1-√5/2 | -1+√5/2 | 1 | 1+√5/2 | 1-√5/2 | i | -i | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ4 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | -1+√5/2 | -1-√5/2 | 1 | 1-√5/2 | 1+√5/2 | i | -i | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ5 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | -1+√5/2 | -1-√5/2 | 1 | 1-√5/2 | 1+√5/2 | -i | i | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ6 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | -1-√5/2 | -1+√5/2 | 1 | 1+√5/2 | 1-√5/2 | -i | i | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ7 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ8 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ9 | 3 | 3 | 1 | 0 | -3 | -3 | -1 | 1+√5/2 | 1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2×A5 |
| ρ10 | 3 | 3 | 1 | 0 | -3 | -3 | -1 | 1-√5/2 | 1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2×A5 |
| ρ11 | 4 | 4 | 0 | 1 | -4 | -4 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ12 | 4 | 4 | 0 | 1 | 4 | 4 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ13 | 4 | -4 | 0 | 1 | 4i | -4i | 0 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | -i | complex faithful |
| ρ14 | 4 | -4 | 0 | 1 | -4i | 4i | 0 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | i | complex faithful |
| ρ15 | 5 | 5 | -1 | -1 | -5 | -5 | 1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ16 | 5 | 5 | 1 | -1 | 5 | 5 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ17 | 6 | -6 | 0 | 0 | 6i | -6i | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | i | -i | -i | i | complex faithful |
| ρ18 | 6 | -6 | 0 | 0 | -6i | 6i | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | -i | i | i | -i | complex faithful |
►On 24 points - transitive group 24T576
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 8)(2 12)(3 15)(4 23)(5 18)(6 14)(7 22)(10 20)(11 19)(17 21)
G:=sub<Sym(24)| (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,8)(2,12)(3,15)(4,23)(5,18)(6,14)(7,22)(10,20)(11,19)(17,21)>;
G:=Group( (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,8)(2,12)(3,15)(4,23)(5,18)(6,14)(7,22)(10,20)(11,19)(17,21) );
G=PermutationGroup([[(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19),(20,21,22,23,24)], [(1,8),(2,12),(3,15),(4,23),(5,18),(6,14),(7,22),(10,20),(11,19),(17,21)]])
G:=TransitiveGroup(24,576);
C4.A5 is a maximal subgroup of
GL2(𝔽5) C8.A5 C4.6S5 C4.S5 C4.3S5 D4.A5 Q8.A5
C4.A5 is a maximal quotient of C4×SL2(𝔽5)
Matrix representation of C4.A5 ►in GL2(𝔽5) generated by
| 4 | 3 |
| 2 | 3 |
| 3 | 1 |
| 2 | 2 |
G:=sub<GL(2,GF(5))| [4,2,3,3],[3,2,1,2] >;
C4.A5 in GAP, Magma, Sage, TeX
C_4.A_5
% in TeX
G:=Group("C4.A5"); // GroupNames label
G:=SmallGroup(240,93);
// by ID
G=gap.SmallGroup(240,93);
# by ID
Export
Subgroup lattice of C4.A5 in TeX
Character table of C4.A5 in TeX