direct product, non-abelian, not soluble, A-group
Aliases: C6×A5, SmallGroup(360,122)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C6×A5 |
| A5 — C6×A5 |
15C2
15C2
10C3
20C3
6C5
5C22
15C22
15C22
10S3
10S3
10C6
15C6
15C6
20C6
10C32
6D5
6C10
6D5
6C15
5C23
5A4
5A4
5A4
10D6
15C2×C6
15C2×C6
10C3×C6
10C3×S3
10C3×S3
6D10
6C30
10S3×C6
Character table of C6×A5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 10A | 10B | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 15 | 15 | 1 | 1 | 20 | 20 | 20 | 12 | 12 | 1 | 1 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 12 | 12 | 12 | 12 | 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 | 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 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ6 | -1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ65 | -1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 3 | -3 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | -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 | orthogonal lifted from C2×A5 |
| ρ8 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 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 | orthogonal lifted from A5 |
| ρ9 | 3 | -3 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | -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 | orthogonal lifted from C2×A5 |
| ρ10 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 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 | orthogonal lifted from A5 |
| ρ11 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | complex lifted from C3×A5 |
| ρ12 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ3ζ53-ζ3ζ52 | complex lifted from C3×A5 |
| ρ13 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | complex lifted from C3×A5 |
| ρ14 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ32ζ54-ζ32ζ5 | complex lifted from C3×A5 |
| ρ15 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ6 | ζ65 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | complex faithful |
| ρ16 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ6 | ζ65 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ53-ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | complex faithful |
| ρ17 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ65 | ζ6 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | complex faithful |
| ρ18 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ65 | ζ6 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ3ζ54-ζ3ζ5 | -ζ32ζ53-ζ32ζ52 | -ζ3ζ53-ζ3ζ52 | -ζ32ζ54-ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 4 | 1 | 1 | 1 | -1 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ20 | 4 | 4 | 0 | 0 | 4 | 4 | 1 | 1 | 1 | -1 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ21 | 4 | 4 | 0 | 0 | -2-2√-3 | -2+2√-3 | ζ32 | ζ3 | 1 | -1 | -1 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 1 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×A5 |
| ρ22 | 4 | -4 | 0 | 0 | -2-2√-3 | -2+2√-3 | ζ32 | ζ3 | 1 | -1 | -1 | 2+2√-3 | 2-2√-3 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | -2+2√-3 | -2-2√-3 | ζ3 | ζ32 | 1 | -1 | -1 | 2-2√-3 | 2+2√-3 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | complex faithful |
| ρ24 | 4 | 4 | 0 | 0 | -2+2√-3 | -2-2√-3 | ζ3 | ζ32 | 1 | -1 | -1 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 1 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×A5 |
| ρ25 | 5 | -5 | -1 | 1 | 5 | 5 | -1 | -1 | -1 | 0 | 0 | -5 | -5 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ26 | 5 | 5 | 1 | 1 | 5 | 5 | -1 | -1 | -1 | 0 | 0 | 5 | 5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ27 | 5 | 5 | 1 | 1 | -5+5√-3/2 | -5-5√-3/2 | ζ65 | ζ6 | -1 | 0 | 0 | -5+5√-3/2 | -5-5√-3/2 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A5 |
| ρ28 | 5 | -5 | -1 | 1 | -5-5√-3/2 | -5+5√-3/2 | ζ6 | ζ65 | -1 | 0 | 0 | 5+5√-3/2 | 5-5√-3/2 | ζ65 | ζ6 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 5 | -5 | -1 | 1 | -5+5√-3/2 | -5-5√-3/2 | ζ65 | ζ6 | -1 | 0 | 0 | 5-5√-3/2 | 5+5√-3/2 | ζ6 | ζ65 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 5 | 5 | 1 | 1 | -5-5√-3/2 | -5+5√-3/2 | ζ6 | ζ65 | -1 | 0 | 0 | -5-5√-3/2 | -5+5√-3/2 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×A5 |
►On 30 points - transitive group 30T87
Generators in S30
(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)
(1 3 20 2 4 23)(5 17 7 6 14 8)(9 12 29 10 11 26)(13 28 22 16 25 19)(15 30 24 18 27 21)
G:=sub<Sym(30)| (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), (1,3,20,2,4,23)(5,17,7,6,14,8)(9,12,29,10,11,26)(13,28,22,16,25,19)(15,30,24,18,27,21)>;
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), (1,3,20,2,4,23)(5,17,7,6,14,8)(9,12,29,10,11,26)(13,28,22,16,25,19)(15,30,24,18,27,21) );
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)], [(1,3,20,2,4,23),(5,17,7,6,14,8),(9,12,29,10,11,26),(13,28,22,16,25,19),(15,30,24,18,27,21)]])
G:=TransitiveGroup(30,87);
►On 30 points - transitive group 30T92
Generators in S30
(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)
(1 26 19 4 29 22)(2 25 9 5 28 12)(3 30 16 6 27 13)(7 23 17 10 20 14)(8 11)(15 18)(21 24)
G:=sub<Sym(30)| (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), (1,26,19,4,29,22)(2,25,9,5,28,12)(3,30,16,6,27,13)(7,23,17,10,20,14)(8,11)(15,18)(21,24)>;
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), (1,26,19,4,29,22)(2,25,9,5,28,12)(3,30,16,6,27,13)(7,23,17,10,20,14)(8,11)(15,18)(21,24) );
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)], [(1,26,19,4,29,22),(2,25,9,5,28,12),(3,30,16,6,27,13),(7,23,17,10,20,14),(8,11),(15,18),(21,24)]])
G:=TransitiveGroup(30,92);
Matrix representation of C6×A5 ►in GL4(𝔽7) generated by
| 4 | 5 | 2 | 6 |
| 6 | 6 | 6 | 5 |
| 5 | 2 | 0 | 0 |
| 6 | 3 | 0 | 3 |
| 1 | 6 | 0 | 6 |
| 3 | 3 | 2 | 6 |
| 2 | 3 | 5 | 2 |
| 5 | 4 | 5 | 1 |
G:=sub<GL(4,GF(7))| [4,6,5,6,5,6,2,3,2,6,0,0,6,5,0,3],[1,3,2,5,6,3,3,4,0,2,5,5,6,6,2,1] >;
C6×A5 in GAP, Magma, Sage, TeX
C_6\times A_5
% in TeX
G:=Group("C6xA5"); // GroupNames label
G:=SmallGroup(360,122);
// by ID
G=gap.SmallGroup(360,122);
# by ID
Export
Subgroup lattice of C6×A5 in TeX
Character table of C6×A5 in TeX