direct product, non-abelian, not soluble, A-group
Aliases: C5×A5, U2(𝔽4), SmallGroup(300,22)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C5×A5 |
| A5 — C5×A5 |
Character table of C5×A5
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 5I | 5J | 5K | 5L | 5M | 5N | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | |
| size | 1 | 15 | 20 | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ5 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | linear of order 5 |
| ρ3 | 1 | 1 | 1 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | ζ53 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | linear of order 5 |
| ρ4 | 1 | 1 | 1 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ52 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | linear of order 5 |
| ρ5 | 1 | 1 | 1 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ54 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | linear of order 5 |
| ρ6 | 3 | -1 | 0 | 3 | 3 | 3 | 3 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ7 | 3 | -1 | 0 | 3 | 3 | 3 | 3 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ8 | 3 | -1 | 0 | 3ζ5 | 3ζ53 | 3ζ52 | 3ζ54 | -ζ54-ζ53 | -ζ53-ζ5 | -ζ54-ζ52 | -ζ52-ζ5 | 1-√5/2 | 1+√5/2 | -ζ52-1 | -ζ54-1 | -ζ5-1 | -ζ53-1 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ9 | 3 | -1 | 0 | 3ζ52 | 3ζ5 | 3ζ54 | 3ζ53 | -ζ53-ζ5 | -ζ52-ζ5 | -ζ54-ζ53 | -ζ54-ζ52 | 1+√5/2 | 1-√5/2 | -ζ54-1 | -ζ53-1 | -ζ52-1 | -ζ5-1 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | 0 | 0 | 0 | 0 | complex faithful |
| ρ10 | 3 | -1 | 0 | 3ζ54 | 3ζ52 | 3ζ53 | 3ζ5 | -ζ52-ζ5 | -ζ54-ζ52 | -ζ53-ζ5 | -ζ54-ζ53 | 1-√5/2 | 1+√5/2 | -ζ53-1 | -ζ5-1 | -ζ54-1 | -ζ52-1 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 3 | -1 | 0 | 3ζ54 | 3ζ52 | 3ζ53 | 3ζ5 | -ζ53-1 | -ζ5-1 | -ζ54-1 | -ζ52-1 | 1+√5/2 | 1-√5/2 | -ζ52-ζ5 | -ζ54-ζ52 | -ζ53-ζ5 | -ζ54-ζ53 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | 0 | 0 | 0 | 0 | complex faithful |
| ρ12 | 3 | -1 | 0 | 3ζ52 | 3ζ5 | 3ζ54 | 3ζ53 | -ζ54-1 | -ζ53-1 | -ζ52-1 | -ζ5-1 | 1-√5/2 | 1+√5/2 | -ζ53-ζ5 | -ζ52-ζ5 | -ζ54-ζ53 | -ζ54-ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 3 | -1 | 0 | 3ζ5 | 3ζ53 | 3ζ52 | 3ζ54 | -ζ52-1 | -ζ54-1 | -ζ5-1 | -ζ53-1 | 1+√5/2 | 1-√5/2 | -ζ54-ζ53 | -ζ53-ζ5 | -ζ54-ζ52 | -ζ52-ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 3 | -1 | 0 | 3ζ53 | 3ζ54 | 3ζ5 | 3ζ52 | -ζ54-ζ52 | -ζ54-ζ53 | -ζ52-ζ5 | -ζ53-ζ5 | 1+√5/2 | 1-√5/2 | -ζ5-1 | -ζ52-1 | -ζ53-1 | -ζ54-1 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 3 | -1 | 0 | 3ζ53 | 3ζ54 | 3ζ5 | 3ζ52 | -ζ5-1 | -ζ52-1 | -ζ53-1 | -ζ54-1 | 1-√5/2 | 1+√5/2 | -ζ54-ζ52 | -ζ54-ζ53 | -ζ52-ζ5 | -ζ53-ζ5 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 4 | 0 | 1 | 4 | 4 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from A5 |
| ρ17 | 4 | 0 | 1 | 4ζ52 | 4ζ5 | 4ζ54 | 4ζ53 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | -1 | -1 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | 0 | 0 | 0 | 0 | ζ54 | ζ53 | ζ5 | ζ52 | complex faithful |
| ρ18 | 4 | 0 | 1 | 4ζ5 | 4ζ53 | 4ζ52 | 4ζ54 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | -1 | -1 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | 0 | 0 | 0 | 0 | ζ52 | ζ54 | ζ53 | ζ5 | complex faithful |
| ρ19 | 4 | 0 | 1 | 4ζ54 | 4ζ52 | 4ζ53 | 4ζ5 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | -1 | -1 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | 0 | 0 | 0 | 0 | ζ53 | ζ5 | ζ52 | ζ54 | complex faithful |
| ρ20 | 4 | 0 | 1 | 4ζ53 | 4ζ54 | 4ζ5 | 4ζ52 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | -1 | -1 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | 0 | 0 | 0 | 0 | ζ5 | ζ52 | ζ54 | ζ53 | complex faithful |
| ρ21 | 5 | 1 | -1 | 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ22 | 5 | 1 | -1 | 5ζ52 | 5ζ5 | 5ζ54 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53 | ζ54 | ζ5 | ζ52 | -ζ54 | -ζ53 | -ζ5 | -ζ52 | complex faithful |
| ρ23 | 5 | 1 | -1 | 5ζ54 | 5ζ52 | 5ζ53 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ5 | ζ53 | ζ52 | ζ54 | -ζ53 | -ζ5 | -ζ52 | -ζ54 | complex faithful |
| ρ24 | 5 | 1 | -1 | 5ζ53 | 5ζ54 | 5ζ5 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ52 | ζ5 | ζ54 | ζ53 | -ζ5 | -ζ52 | -ζ54 | -ζ53 | complex faithful |
| ρ25 | 5 | 1 | -1 | 5ζ5 | 5ζ53 | 5ζ52 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54 | ζ52 | ζ53 | ζ5 | -ζ52 | -ζ54 | -ζ53 | -ζ5 | complex faithful |
►On 25 points - transitive group 25T29
Generators in S25
(1 9 23 21 19)(2 10 14 12 25)(3 6 20 18 16)(4 7 11 24 22)(5 8 17 15 13)
(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)
G:=sub<Sym(25)| (1,9,23,21,19)(2,10,14,12,25)(3,6,20,18,16)(4,7,11,24,22)(5,8,17,15,13), (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)>;
G:=Group( (1,9,23,21,19)(2,10,14,12,25)(3,6,20,18,16)(4,7,11,24,22)(5,8,17,15,13), (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) );
G=PermutationGroup([[(1,9,23,21,19),(2,10,14,12,25),(3,6,20,18,16),(4,7,11,24,22),(5,8,17,15,13)], [(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)]])
G:=TransitiveGroup(25,29);
►On 30 points - transitive group 30T69
Generators in S30
(1 9 17 5 18)(2 30 13 6 29)(3 26 14 27 10)(4 12 20 8 21)(7 15 23 11 24)(16 19 22 25 28)
(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)
G:=sub<Sym(30)| (1,9,17,5,18)(2,30,13,6,29)(3,26,14,27,10)(4,12,20,8,21)(7,15,23,11,24)(16,19,22,25,28), (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)>;
G:=Group( (1,9,17,5,18)(2,30,13,6,29)(3,26,14,27,10)(4,12,20,8,21)(7,15,23,11,24)(16,19,22,25,28), (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) );
G=PermutationGroup([[(1,9,17,5,18),(2,30,13,6,29),(3,26,14,27,10),(4,12,20,8,21),(7,15,23,11,24),(16,19,22,25,28)], [(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)]])
G:=TransitiveGroup(30,69);
Matrix representation of C5×A5 ►in GL3(𝔽11) generated by
| 10 | 9 | 0 |
| 3 | 3 | 0 |
| 3 | 8 | 1 |
| 0 | 6 | 9 |
| 0 | 1 | 4 |
| 5 | 5 | 10 |
G:=sub<GL(3,GF(11))| [10,3,3,9,3,8,0,0,1],[0,0,5,6,1,5,9,4,10] >;
C5×A5 in GAP, Magma, Sage, TeX
C_5\times A_5
% in TeX
G:=Group("C5xA5"); // GroupNames label
G:=SmallGroup(300,22);
// by ID
G=gap.SmallGroup(300,22);
# by ID
Export
Subgroup lattice of C5×A5 in TeX
Character table of C5×A5 in TeX