direct product, non-abelian, not soluble, A-group
Aliases: S3×A5, C3⋊(C2×A5), (C3×A5)⋊1C2, SmallGroup(360,121)
Series: Chief►Derived ►Lower central ►Upper central
| C3×A5 — S3×A5 |
3C2
15C2
45C2
10C3
20C3
6C5
5C22
45C22
45C22
10S3
15C6
15S3
30S3
30C6
60S3
10C32
6D5
18D5
18C10
6C15
15C23
5A4
10A4
15D6
15D6
30D6
10C3×S3
10C3×S3
10C3⋊S3
18D10
6D15
15C2×A4
10S32
Character table of S3×A5
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 5A | 5B | 6A | 6B | 10A | 10B | 15A | 15B | |
| size | 1 | 3 | 15 | 45 | 2 | 20 | 40 | 12 | 12 | 30 | 60 | 36 | 36 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 0 | 2 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 3 | -3 | -1 | 1 | 3 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2×A5 |
| ρ5 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ6 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ7 | 3 | -3 | -1 | 1 | 3 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2×A5 |
| ρ8 | 4 | -4 | 0 | 0 | 4 | 1 | 1 | -1 | -1 | 0 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A5 |
| ρ9 | 4 | 4 | 0 | 0 | 4 | 1 | 1 | -1 | -1 | 0 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ10 | 5 | -5 | 1 | -1 | 5 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ11 | 5 | 5 | 1 | 1 | 5 | -1 | -1 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ12 | 6 | 0 | -2 | 0 | -3 | 0 | 0 | 1-√5 | 1+√5 | 1 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ13 | 6 | 0 | -2 | 0 | -3 | 0 | 0 | 1+√5 | 1-√5 | 1 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ14 | 8 | 0 | 0 | 0 | -4 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal faithful |
| ρ15 | 10 | 0 | 2 | 0 | -5 | -2 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 15 points - transitive group 15T23
Generators in S15
(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15)
(1 13 8)(2 15 12)(3 7 14)(4 11 6)(5 9 10)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,13,8)(2,15,12)(3,7,14)(4,11,6)(5,9,10)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,13,8)(2,15,12)(3,7,14)(4,11,6)(5,9,10) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10,11,12,13,14,15)], [(1,13,8),(2,15,12),(3,7,14),(4,11,6),(5,9,10)]])
G:=TransitiveGroup(15,23);
►On 18 points - transitive group 18T145
Generators in S18
(2 3)(4 5 6 7 8)(9 10 11 12 13 14 15 16 17 18)
(1 14 13)(2 4 18)(3 9 8)(5 16 17)(6 12 15)(7 10 11)
G:=sub<Sym(18)| (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,14,13)(2,4,18)(3,9,8)(5,16,17)(6,12,15)(7,10,11)>;
G:=Group( (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,14,13)(2,4,18)(3,9,8)(5,16,17)(6,12,15)(7,10,11) );
G=PermutationGroup([[(2,3),(4,5,6,7,8),(9,10,11,12,13,14,15,16,17,18)], [(1,14,13),(2,4,18),(3,9,8),(5,16,17),(6,12,15),(7,10,11)]])
G:=TransitiveGroup(18,145);
►On 30 points - transitive group 30T85
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 16 25)(2 22 15)(3 18 21)(4 24 17)(5 14 23)(6 28 13)(7 20 27)(8 30 19)(9 12 29)(10 26 11)
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), (1,16,25)(2,22,15)(3,18,21)(4,24,17)(5,14,23)(6,28,13)(7,20,27)(8,30,19)(9,12,29)(10,26,11)>;
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), (1,16,25)(2,22,15)(3,18,21)(4,24,17)(5,14,23)(6,28,13)(7,20,27)(8,30,19)(9,12,29)(10,26,11) );
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)], [(1,16,25),(2,22,15),(3,18,21),(4,24,17),(5,14,23),(6,28,13),(7,20,27),(8,30,19),(9,12,29),(10,26,11)]])
G:=TransitiveGroup(30,85);
►On 30 points - transitive group 30T94
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 14)(2 25 30)(3 19 24)(4 17 18)(5 15 16)(6 29 26)(7 11 28)(8 23 20)(9 13 22)(10 27 12)
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), (1,21,14)(2,25,30)(3,19,24)(4,17,18)(5,15,16)(6,29,26)(7,11,28)(8,23,20)(9,13,22)(10,27,12)>;
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), (1,21,14)(2,25,30)(3,19,24)(4,17,18)(5,15,16)(6,29,26)(7,11,28)(8,23,20)(9,13,22)(10,27,12) );
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)], [(1,21,14),(2,25,30),(3,19,24),(4,17,18),(5,15,16),(6,29,26),(7,11,28),(8,23,20),(9,13,22),(10,27,12)]])
G:=TransitiveGroup(30,94);
►On 30 points - transitive group 30T102
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 29)(4 13 28)(5 20 12)(6 17 19)(7 21 16)(8 10 30)(14 25 27)(15 22 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), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24)>;
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), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24) );
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)], [(1,3,29),(4,13,28),(5,20,12),(6,17,19),(7,21,16),(8,10,30),(14,25,27),(15,22,24)]])
G:=TransitiveGroup(30,102);
Polynomial with Galois group S3×A5 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T23 | x15+5x14+8x13+7x12+8x11-3x9+7x8-2x7+7x6-2x5-3x4+5x3-2x+1 | -237·416·4632 |
Matrix representation of S3×A5 ►in GL5(𝔽31)
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 29 | 21 |
| 0 | 0 | 2 | 10 | 3 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 |
| 1 | 30 | 0 | 0 | 0 |
| 0 | 0 | 28 | 21 | 29 |
| 0 | 0 | 29 | 3 | 10 |
| 0 | 0 | 30 | 0 | 0 |
G:=sub<GL(5,GF(31))| [0,1,0,0,0,1,0,0,0,0,0,0,3,2,1,0,0,29,10,0,0,0,21,3,0],[0,1,0,0,0,30,30,0,0,0,0,0,28,29,30,0,0,21,3,0,0,0,29,10,0] >;
S3×A5 in GAP, Magma, Sage, TeX
S_3\times A_5
% in TeX
G:=Group("S3xA5"); // GroupNames label
G:=SmallGroup(360,121);
// by ID
G=gap.SmallGroup(360,121);
# by ID
Export
Subgroup lattice of S3×A5 in TeX
Character table of S3×A5 in TeX