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## G = C2×S5order 240 = 24·3·5

### Direct product of C2 and S5

Aliases: C2×S5, O3(𝔽5), A5⋊C22, (C2×A5)⋊C2, SmallGroup(240,189)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C2×A5 — C2×S5
 Derived series A5 — C2×S5
 Lower central A5 — C2×S5
 Upper central C1 — C2

Subgroups: 535 in 57 conjugacy classes, 7 normal (5 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C22 [×7], C5, S3 [×4], C6 [×3], C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, A4, D6 [×6], C2×C6, C2×D4, F5 [×2], D10, S4 [×2], C2×A4, C22×S3, C2×F5, C2×S4, A5, S5 [×2], C2×A5, C2×S5
Quotients: C1, C2 [×3], 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

Permutation representations of C2×S5
On 10 points - transitive group 10T22
Generators in S10
(1 8 7 4)(2 9 6 3)(5 10)
(1 2 3 4 5 6 7 8 9 10)

G:=sub<Sym(10)| (1,8,7,4)(2,9,6,3)(5,10), (1,2,3,4,5,6,7,8,9,10)>;

G:=Group( (1,8,7,4)(2,9,6,3)(5,10), (1,2,3,4,5,6,7,8,9,10) );

G=PermutationGroup([(1,8,7,4),(2,9,6,3),(5,10)], [(1,2,3,4,5,6,7,8,9,10)])

G:=TransitiveGroup(10,22);

On 12 points - transitive group 12T123
Generators in S12
(1 11 10 9)(2 6 5 4)(3 8)(7 12)
(1 2)(3 4 5 6 7 8 9 10 11 12)

G:=sub<Sym(12)| (1,11,10,9)(2,6,5,4)(3,8)(7,12), (1,2)(3,4,5,6,7,8,9,10,11,12)>;

G:=Group( (1,11,10,9)(2,6,5,4)(3,8)(7,12), (1,2)(3,4,5,6,7,8,9,10,11,12) );

G=PermutationGroup([(1,11,10,9),(2,6,5,4),(3,8),(7,12)], [(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 19 11 13)(2 10)(3 9 15 17)(4 20 12 8)(5 7)(6 14 16 18)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,19,11,13)(2,10)(3,9,15,17)(4,20,12,8)(5,7)(6,14,16,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,19,11,13)(2,10)(3,9,15,17)(4,20,12,8)(5,7)(6,14,16,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,19,11,13),(2,10),(3,9,15,17),(4,20,12,8),(5,7),(6,14,16,18)], [(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 19 16 13)(2 10 7 5)(3 9 15 17)(4 20 12 8)(6 14 11 18)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,19,16,13)(2,10,7,5)(3,9,15,17)(4,20,12,8)(6,14,11,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,19,16,13)(2,10,7,5)(3,9,15,17)(4,20,12,8)(6,14,11,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,19,16,13),(2,10,7,5),(3,9,15,17),(4,20,12,8),(6,14,11,18)], [(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 13)(2 16 8 12)(3 17 7 11)(4 20 10 14)(5 19 9 15)(6 18)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,13)(2,16,8,12)(3,17,7,11)(4,20,10,14)(5,19,9,15)(6,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,13)(2,16,8,12)(3,17,7,11)(4,20,10,14)(5,19,9,15)(6,18), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,13),(2,16,8,12),(3,17,7,11),(4,20,10,14),(5,19,9,15),(6,18)], [(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 9 21 7)(2 14 16 12)(3 22 13 20)(4 17 8 15)(5 23 10 18)(6 19 11 24)
(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)| (1,9,21,7)(2,14,16,12)(3,22,13,20)(4,17,8,15)(5,23,10,18)(6,19,11,24), (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( (1,9,21,7)(2,14,16,12)(3,22,13,20)(4,17,8,15)(5,23,10,18)(6,19,11,24), (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([(1,9,21,7),(2,14,16,12),(3,22,13,20),(4,17,8,15),(5,23,10,18),(6,19,11,24)], [(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 17 6 15)(2 22 11 20)(3 12 21 10)(4 7 16 5)(8 23)(9 24)(13 18)(14 19)
(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)| (1,17,6,15)(2,22,11,20)(3,12,21,10)(4,7,16,5)(8,23)(9,24)(13,18)(14,19), (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( (1,17,6,15)(2,22,11,20)(3,12,21,10)(4,7,16,5)(8,23)(9,24)(13,18)(14,19), (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([(1,17,6,15),(2,22,11,20),(3,12,21,10),(4,7,16,5),(8,23),(9,24),(13,18),(14,19)], [(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 12 29 26)(2 18 15 8)(3 7 13 20)(4 23 30 16)(6 17 24 21)(9 28 25 11)(14 22 19 27)
(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)| (1,12,29,26)(2,18,15,8)(3,7,13,20)(4,23,30,16)(6,17,24,21)(9,28,25,11)(14,22,19,27), (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( (1,12,29,26)(2,18,15,8)(3,7,13,20)(4,23,30,16)(6,17,24,21)(9,28,25,11)(14,22,19,27), (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([(1,12,29,26),(2,18,15,8),(3,7,13,20),(4,23,30,16),(6,17,24,21),(9,28,25,11),(14,22,19,27)], [(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 25 18 20)(2 26 28 8)(3 7 21 23)(4 17 19 24)(6 30 13 15)(9 12 14 29)(11 22)(16 27)
(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)| (1,25,18,20)(2,26,28,8)(3,7,21,23)(4,17,19,24)(6,30,13,15)(9,12,14,29)(11,22)(16,27), (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( (1,25,18,20)(2,26,28,8)(3,7,21,23)(4,17,19,24)(6,30,13,15)(9,12,14,29)(11,22)(16,27), (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([(1,25,18,20),(2,26,28,8),(3,7,21,23),(4,17,19,24),(6,30,13,15),(9,12,14,29),(11,22),(16,27)], [(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

Polynomial with Galois group C2×S5 over ℚ
actionf(x)Disc(f)
10T22x10-15x8-4x7+46x6+16x5-46x4-17x3+12x2+3x-155·612·1072·3972
12T123x12-14x10+71x8-156x6+135x4-28x2+1224·36·74·236

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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