direct product, non-abelian, not soluble
Aliases: C3×S5, A5⋊C6, (C3×A5)⋊3C2, SmallGroup(360,119)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C3×S5 |
| A5 — C3×S5 |
10C2
15C2
10C3
20C3
6C5
5C22
15C22
15C4
10S3
10C6
10C6
10S3
15C6
20C6
10C32
6D5
6C15
15D4
5A4
10D6
10A4
15C2×C6
15C12
10C3×S3
10C3×C6
10C3×S3
6F5
5S4
15C3×D4
10S3×C6
Character table of C3×S5
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4 | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 15A | 15B | |
| size | 1 | 10 | 15 | 1 | 1 | 20 | 20 | 20 | 30 | 24 | 10 | 10 | 15 | 15 | 20 | 20 | 20 | 30 | 30 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | 1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | 1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 4 | 2 | 0 | 4 | 4 | 1 | 1 | 1 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | -2 | 0 | 4 | 4 | 1 | 1 | 1 | 0 | -1 | -2 | -2 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from S5 |
| ρ9 | 4 | 2 | 0 | -2-2√-3 | -2+2√-3 | ζ3 | ζ32 | 1 | 0 | -1 | -1+√-3 | -1-√-3 | 0 | 0 | ζ6 | ζ65 | -1 | 0 | 0 | ζ65 | ζ6 | complex faithful |
| ρ10 | 4 | 2 | 0 | -2+2√-3 | -2-2√-3 | ζ32 | ζ3 | 1 | 0 | -1 | -1-√-3 | -1+√-3 | 0 | 0 | ζ65 | ζ6 | -1 | 0 | 0 | ζ6 | ζ65 | complex faithful |
| ρ11 | 4 | -2 | 0 | -2+2√-3 | -2-2√-3 | ζ32 | ζ3 | 1 | 0 | -1 | 1+√-3 | 1-√-3 | 0 | 0 | ζ3 | ζ32 | 1 | 0 | 0 | ζ6 | ζ65 | complex faithful |
| ρ12 | 4 | -2 | 0 | -2-2√-3 | -2+2√-3 | ζ3 | ζ32 | 1 | 0 | -1 | 1-√-3 | 1+√-3 | 0 | 0 | ζ32 | ζ3 | 1 | 0 | 0 | ζ65 | ζ6 | complex faithful |
| ρ13 | 5 | 1 | 1 | 5 | 5 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ14 | 5 | -1 | 1 | 5 | 5 | -1 | -1 | -1 | 1 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from S5 |
| ρ15 | 5 | -1 | 1 | -5-5√-3/2 | -5+5√-3/2 | ζ65 | ζ6 | -1 | 1 | 0 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | -1 | ζ32 | ζ3 | 0 | 0 | complex faithful |
| ρ16 | 5 | 1 | 1 | -5-5√-3/2 | -5+5√-3/2 | ζ65 | ζ6 | -1 | -1 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | 0 | 0 | complex faithful |
| ρ17 | 5 | 1 | 1 | -5+5√-3/2 | -5-5√-3/2 | ζ6 | ζ65 | -1 | -1 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | 0 | 0 | complex faithful |
| ρ18 | 5 | -1 | 1 | -5+5√-3/2 | -5-5√-3/2 | ζ6 | ζ65 | -1 | 1 | 0 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | -1 | ζ3 | ζ32 | 0 | 0 | complex faithful |
| ρ19 | 6 | 0 | -2 | 6 | 6 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ20 | 6 | 0 | -2 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex faithful |
| ρ21 | 6 | 0 | -2 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex faithful |
►On 15 points - transitive group 15T24
Generators in S15
(1 2 3)(4 5 6)(7 8 9)(10 11 12 13 14 15)
(2 5 12)(3 10 8)(4 9 14)
G:=sub<Sym(15)| (1,2,3)(4,5,6)(7,8,9)(10,11,12,13,14,15), (2,5,12)(3,10,8)(4,9,14)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12,13,14,15), (2,5,12)(3,10,8)(4,9,14) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12,13,14,15)], [(2,5,12),(3,10,8),(4,9,14)]])
G:=TransitiveGroup(15,24);
►On 18 points - transitive group 18T144
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 14 17)(2 15 5)(3 6 10)(4 8 11)(7 16 13)(9 18 12)
G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,14,17)(2,15,5)(3,6,10)(4,8,11)(7,16,13)(9,18,12)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,14,17)(2,15,5)(3,6,10)(4,8,11)(7,16,13)(9,18,12) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,14,17),(2,15,5),(3,6,10),(4,8,11),(7,16,13),(9,18,12)]])
G:=TransitiveGroup(18,144);
►On 30 points - transitive group 30T90
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 4 5)(2 23 7)(3 9 29)(6 25 21)(8 22 30)(10 17 12)(11 27 19)(13 28 26)(14 18 16)(15 20 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), (1,4,5)(2,23,7)(3,9,29)(6,25,21)(8,22,30)(10,17,12)(11,27,19)(13,28,26)(14,18,16)(15,20,24)>;
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), (1,4,5)(2,23,7)(3,9,29)(6,25,21)(8,22,30)(10,17,12)(11,27,19)(13,28,26)(14,18,16)(15,20,24) );
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)], [(1,4,5),(2,23,7),(3,9,29),(6,25,21),(8,22,30),(10,17,12),(11,27,19),(13,28,26),(14,18,16),(15,20,24)]])
G:=TransitiveGroup(30,90);
►On 30 points - transitive group 30T98
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 21 18)(2 27 19)(3 14 25)(4 28 17)(5 26 9)(6 11 13)(7 22 30)(8 20 12)(10 15 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), (1,21,18)(2,27,19)(3,14,25)(4,28,17)(5,26,9)(6,11,13)(7,22,30)(8,20,12)(10,15,24)>;
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), (1,21,18)(2,27,19)(3,14,25)(4,28,17)(5,26,9)(6,11,13)(7,22,30)(8,20,12)(10,15,24) );
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)], [(1,21,18),(2,27,19),(3,14,25),(4,28,17),(5,26,9),(6,11,13),(7,22,30),(8,20,12),(10,15,24)]])
G:=TransitiveGroup(30,98);
►On 30 points - transitive group 30T103
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 14)(2 20 9)(4 29 22)(5 18 12)(7 24 25)(10 16 28)
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), (1,26,14)(2,20,9)(4,29,22)(5,18,12)(7,24,25)(10,16,28)>;
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), (1,26,14)(2,20,9)(4,29,22)(5,18,12)(7,24,25)(10,16,28) );
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)], [(1,26,14),(2,20,9),(4,29,22),(5,18,12),(7,24,25),(10,16,28)]])
G:=TransitiveGroup(30,103);
Polynomial with Galois group C3×S5 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T24 | x15-x14-2x13+x12+16x10+22x9-13x8-57x5+26x4-1 | 212·710·193·1132·1513·19732·124332 |
Matrix representation of C3×S5 ►in GL4(𝔽7) generated by
| 4 | 3 | 2 | 5 |
| 0 | 0 | 3 | 5 |
| 1 | 3 | 2 | 0 |
| 4 | 0 | 0 | 5 |
| 0 | 3 | 0 | 4 |
| 2 | 0 | 5 | 1 |
| 5 | 3 | 5 | 4 |
| 2 | 5 | 4 | 3 |
G:=sub<GL(4,GF(7))| [4,0,1,4,3,0,3,0,2,3,2,0,5,5,0,5],[0,2,5,2,3,0,3,5,0,5,5,4,4,1,4,3] >;
C3×S5 in GAP, Magma, Sage, TeX
C_3\times S_5
% in TeX
G:=Group("C3xS5"); // GroupNames label
G:=SmallGroup(360,119);
// by ID
G=gap.SmallGroup(360,119);
# by ID
Export
Subgroup lattice of C3×S5 in TeX
Character table of C3×S5 in TeX