direct product, non-abelian, not soluble, A-group
Aliases: C4×A5, C2.1(C2×A5), (C2×A5).2C2, SmallGroup(240,92)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C4×A5 |
| A5 — C4×A5 |
15C2
15C2
10C3
6C5
5C22
15C4
15C22
15C22
10S3
10C6
10S3
6C10
6D5
6D5
5C23
15C2×C4
15C2×C4
5A4
10C12
10D6
10Dic3
6Dic5
6C20
6D10
10C4×S3
Character table of C4×A5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 15 | 15 | 20 | 1 | 1 | 15 | 15 | 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 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 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 | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | linear of order 4 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | linear of order 4 |
| ρ5 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -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 |
| ρ6 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -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 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 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 |
| ρ8 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 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 |
| ρ9 | 3 | -3 | 1 | -1 | 0 | 3i | -3i | -i | i | 1-√5/2 | 1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ10 | 3 | -3 | 1 | -1 | 0 | -3i | 3i | i | -i | 1+√5/2 | 1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ11 | 3 | -3 | 1 | -1 | 0 | 3i | -3i | -i | i | 1+√5/2 | 1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ12 | 3 | -3 | 1 | -1 | 0 | -3i | 3i | i | -i | 1-√5/2 | 1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ13 | 4 | 4 | 0 | 0 | 1 | -4 | -4 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ14 | 4 | 4 | 0 | 0 | 1 | 4 | 4 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ15 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | -i | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | complex faithful |
| ρ17 | 5 | 5 | 1 | 1 | -1 | -5 | -5 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ18 | 5 | 5 | 1 | 1 | -1 | 5 | 5 | 1 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ19 | 5 | -5 | -1 | 1 | -1 | -5i | 5i | -i | i | 0 | 0 | 1 | 0 | 0 | -i | i | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 5 | -5 | -1 | 1 | -1 | 5i | -5i | i | -i | 0 | 0 | 1 | 0 | 0 | i | -i | 0 | 0 | 0 | 0 | complex faithful |
►On 20 points - transitive group 20T63
Generators in S20
(1 14 11 4)(2 7 12 17)(3 20 13 10)(5 18 15 8)(6 19 16 9)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,14,11,4)(2,7,12,17)(3,20,13,10)(5,18,15,8)(6,19,16,9), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,14,11,4)(2,7,12,17)(3,20,13,10)(5,18,15,8)(6,19,16,9), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,14,11,4),(2,7,12,17),(3,20,13,10),(5,18,15,8),(6,19,16,9)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,63);
►On 24 points - transitive group 24T574
Generators in S24
(1 18 3 8)(2 23 4 13)(5 6 15 16)(7 22 17 12)(9 24 19 14)(10 11 20 21)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,18,3,8)(2,23,4,13)(5,6,15,16)(7,22,17,12)(9,24,19,14)(10,11,20,21), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,18,3,8)(2,23,4,13)(5,6,15,16)(7,22,17,12)(9,24,19,14)(10,11,20,21), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,18,3,8),(2,23,4,13),(5,6,15,16),(7,22,17,12),(9,24,19,14),(10,11,20,21)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,574);
►On 24 points - transitive group 24T575
Generators in S24
(1 13 3 23)(2 18 4 8)(5 6 15 16)(7 12 17 22)(9 14 19 24)(10 11 20 21)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,13,3,23)(2,18,4,8)(5,6,15,16)(7,12,17,22)(9,14,19,24)(10,11,20,21), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,13,3,23)(2,18,4,8)(5,6,15,16)(7,12,17,22)(9,14,19,24)(10,11,20,21), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,13,3,23),(2,18,4,8),(5,6,15,16),(7,12,17,22),(9,14,19,24),(10,11,20,21)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,575);
C4×A5 is a maximal subgroup of
A5⋊C8 C4⋊S5 A5⋊Q8
C4×A5 is a maximal quotient of C8.A5
Matrix representation of C4×A5 ►in GL3(𝔽5) generated by
| 4 | 4 | 4 |
| 3 | 0 | 2 |
| 4 | 1 | 4 |
| 3 | 3 | 2 |
| 0 | 4 | 3 |
| 2 | 3 | 4 |
G:=sub<GL(3,GF(5))| [4,3,4,4,0,1,4,2,4],[3,0,2,3,4,3,2,3,4] >;
C4×A5 in GAP, Magma, Sage, TeX
C_4\times A_5
% in TeX
G:=Group("C4xA5"); // GroupNames label
G:=SmallGroup(240,92);
// by ID
G=gap.SmallGroup(240,92);
# by ID
Export
Subgroup lattice of C4×A5 in TeX
Character table of C4×A5 in TeX