direct product, non-abelian, not soluble, rational
Aliases: C2×S5, O3(𝔽5), A5⋊C22, (C2×A5)⋊C2, SmallGroup(240,189)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C2×S5 |
| A5 — C2×S5 |
Subgroups: 535 in 57 conjugacy classes, 7 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C2×C4, D4, C23, D5, C10, A4, D6, C2×C6, C2×D4, F5, D10, S4, C2×A4, C22×S3, C2×F5, C2×S4, A5, S5, C2×A5, C2×S5
Quotients: C1, C2, C22, S5, C2×S5
Character table of C2×S5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5 | 6A | 6B | 6C | 10 | |
| size | 1 | 1 | 10 | 10 | 15 | 15 | 20 | 30 | 30 | 24 | 20 | 20 | 20 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 4 | -4 | -2 | 2 | 0 | 0 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | orthogonal faithful |
| ρ6 | 4 | -4 | 2 | -2 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | orthogonal faithful |
| ρ7 | 4 | 4 | -2 | -2 | 0 | 0 | 1 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | 4 | 2 | 2 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ9 | 5 | -5 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 0 | orthogonal faithful |
| ρ10 | 5 | -5 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 0 | 1 | 1 | -1 | 0 | orthogonal faithful |
| ρ11 | 5 | 5 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | 0 | orthogonal lifted from S5 |
| ρ12 | 5 | 5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 0 | 1 | -1 | 1 | 0 | orthogonal lifted from S5 |
| ρ13 | 6 | 6 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | orthogonal lifted from S5 |
| ρ14 | 6 | -6 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | orthogonal faithful |
►On 10 points - transitive group 10T22
Generators in S10
(1 8 5 4)(2 7)(3 10 9 6)
(1 2 3 4 5 6 7 8 9 10)
G:=sub<Sym(10)| (1,8,5,4)(2,7)(3,10,9,6), (1,2,3,4,5,6,7,8,9,10)>;
G:=Group( (1,8,5,4)(2,7)(3,10,9,6), (1,2,3,4,5,6,7,8,9,10) );
G=PermutationGroup([[(1,8,5,4),(2,7),(3,10,9,6)], [(1,2,3,4,5,6,7,8,9,10)]])
G:=TransitiveGroup(10,22);
►On 12 points - transitive group 12T123
Generators in S12
(1 8 7 6)(2 3 12 11)(4 9)(5 10)
(1 2)(3 4 5 6 7 8 9 10 11 12)
G:=sub<Sym(12)| (1,8,7,6)(2,3,12,11)(4,9)(5,10), (1,2)(3,4,5,6,7,8,9,10,11,12)>;
G:=Group( (1,8,7,6)(2,3,12,11)(4,9)(5,10), (1,2)(3,4,5,6,7,8,9,10,11,12) );
G=PermutationGroup([[(1,8,7,6),(2,3,12,11),(4,9),(5,10)], [(1,2),(3,4,5,6,7,8,9,10,11,12)]])
G:=TransitiveGroup(12,123);
►On 20 points - transitive group 20T62
Generators in S20
(1 3 17 13)(2 4 20 10)(5 7 9 15)(6 8 12 18)(11 19)(14 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,3,17,13)(2,4,20,10)(5,7,9,15)(6,8,12,18)(11,19)(14,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,3,17,13)(2,4,20,10)(5,7,9,15)(6,8,12,18)(11,19)(14,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,3,17,13),(2,4,20,10),(5,7,9,15),(6,8,12,18),(11,19),(14,16)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,62);
►On 20 points - transitive group 20T65
Generators in S20
(1 3 19 15)(2 9 17 5)(4 12 10 7)(6 8 14 20)(11 18 16 13)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,3,19,15)(2,9,17,5)(4,12,10,7)(6,8,14,20)(11,18,16,13), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,3,19,15)(2,9,17,5)(4,12,10,7)(6,8,14,20)(11,18,16,13), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,3,19,15),(2,9,17,5),(4,12,10,7),(6,8,14,20),(11,18,16,13)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,65);
►On 20 points - transitive group 20T70
Generators in S20
(1 15 5 19)(2 18 8 12)(3 17 7 13)(4 16)(6 20 10 14)(9 11)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,15,5,19)(2,18,8,12)(3,17,7,13)(4,16)(6,20,10,14)(9,11), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,15,5,19)(2,18,8,12)(3,17,7,13)(4,16)(6,20,10,14)(9,11), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,15,5,19),(2,18,8,12),(3,17,7,13),(4,16),(6,20,10,14),(9,11)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,70);
►On 24 points - transitive group 24T570
Generators in S24
(1 23 13 21)(2 18 8 16)(3 14 17 12)(4 9 22 7)(5 24 10 19)(6 20 11 15)
(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)| (1,23,13,21)(2,18,8,16)(3,14,17,12)(4,9,22,7)(5,24,10,19)(6,20,11,15), (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( (1,23,13,21)(2,18,8,16)(3,14,17,12)(4,9,22,7)(5,24,10,19)(6,20,11,15), (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([[(1,23,13,21),(2,18,8,16),(3,14,17,12),(4,9,22,7),(5,24,10,19),(6,20,11,15)], [(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(24,570);
►On 24 points - transitive group 24T577
Generators in S24
(1 8 15 6)(2 13 20 11)(3 21 12 19)(4 16 7 24)(5 18)(9 22)(10 23)(14 17)
(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)| (1,8,15,6)(2,13,20,11)(3,21,12,19)(4,16,7,24)(5,18)(9,22)(10,23)(14,17), (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( (1,8,15,6)(2,13,20,11)(3,21,12,19)(4,16,7,24)(5,18)(9,22)(10,23)(14,17), (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([[(1,8,15,6),(2,13,20,11),(3,21,12,19),(4,16,7,24),(5,18),(9,22),(10,23),(14,17)], [(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(24,577);
►On 30 points - transitive group 30T58
Generators in S30
(1 8 23 27)(2 19 7 14)(3 28 22 6)(4 24 20 17)(5 11 18 26)(9 29 15 12)(10 16 13 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 25 26 27 28 29 30)
G:=sub<Sym(30)| (1,8,23,27)(2,19,7,14)(3,28,22,6)(4,24,20,17)(5,11,18,26)(9,29,15,12)(10,16,13,21), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)>;
G:=Group( (1,8,23,27)(2,19,7,14)(3,28,22,6)(4,24,20,17)(5,11,18,26)(9,29,15,12)(10,16,13,21), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30) );
G=PermutationGroup([[(1,8,23,27),(2,19,7,14),(3,28,22,6),(4,24,20,17),(5,11,18,26),(9,29,15,12),(10,16,13,21)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)]])
G:=TransitiveGroup(30,58);
►On 30 points - transitive group 30T60
Generators in S30
(1 3 19 13)(2 23)(4 20 29 21)(5 30 22 17)(6 8 14 18)(7 28)(9 15 24 26)(10 25 27 12)
(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)| (1,3,19,13)(2,23)(4,20,29,21)(5,30,22,17)(6,8,14,18)(7,28)(9,15,24,26)(10,25,27,12), (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( (1,3,19,13)(2,23)(4,20,29,21)(5,30,22,17)(6,8,14,18)(7,28)(9,15,24,26)(10,25,27,12), (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([[(1,3,19,13),(2,23),(4,20,29,21),(5,30,22,17),(6,8,14,18),(7,28),(9,15,24,26),(10,25,27,12)], [(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(30,60);
C2×S5 is a maximal subgroup of
C4⋊S5 C22⋊S5
C2×S5 is a maximal quotient of C4⋊S5 A5⋊Q8 C4.6S5 C4.S5 C4.3S5 C22⋊S5 C22.S5
| action | f(x) | Disc(f) |
|---|---|---|
| 10T22 | x10-15x8-4x7+46x6+16x5-46x4-17x3+12x2+3x-1 | 55·612·1072·3972 |
| 12T123 | x12-14x10+71x8-156x6+135x4-28x2+1 | 224·36·74·236 |
Matrix representation of C2×S5 ►in GL3(𝔽5) generated by
| 2 | 4 | 2 |
| 2 | 0 | 1 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 2 | 0 | 1 |
| 0 | 4 | 3 |
G:=sub<GL(3,GF(5))| [2,2,0,4,0,0,2,1,2],[4,2,0,0,0,4,0,1,3] >;
C2×S5 in GAP, Magma, Sage, TeX
C_2\times S_5
% in TeX
G:=Group("C2xS5"); // GroupNames label
G:=SmallGroup(240,189);
// by ID
G=gap.SmallGroup(240,189);
# by ID
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