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## G = C22×S5order 480 = 25·3·5

### Direct product of C22 and S5

Aliases: C22×S5, A5⋊C23, (C2×A5)⋊C22, (C22×A5)⋊3C2, SmallGroup(480,1186)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C22 — C22×A5 — C22×S5
 Derived series A5 — C22×S5
 Lower central A5 — C22×S5
 Upper central C1 — C22

Subgroups: 2388 in 225 conjugacy classes, 21 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, D5, C10, A4, D6, C2×C6, C22×C4, C2×D4, C24, F5, D10, C2×C10, S4, C2×A4, C22×S3, C22×C6, C22×D4, C2×F5, C22×D5, C2×S4, C22×A4, S3×C23, A5, C22×F5, C22×S4, S5, C2×A5, C2×S5, C22×A5, C22×S5
Quotients: C1, C2, C22, C23, S5, C2×S5, C22×S5

Character table of C22×S5

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C size 1 1 1 1 10 10 10 10 15 15 15 15 20 30 30 30 30 24 20 20 20 20 20 20 20 24 24 24 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 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 linear of order 2 ρ3 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 ρ4 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 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 4 -4 -4 4 -2 2 2 -2 0 0 0 0 1 0 0 0 0 -1 1 -1 -1 1 -1 -1 1 1 -1 1 orthogonal lifted from C2×S5 ρ10 4 4 4 4 -2 -2 -2 -2 0 0 0 0 1 0 0 0 0 -1 1 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from S5 ρ11 4 4 -4 -4 -2 -2 2 2 0 0 0 0 1 0 0 0 0 -1 1 -1 1 -1 1 -1 -1 -1 1 1 orthogonal lifted from C2×S5 ρ12 4 -4 4 -4 -2 2 -2 2 0 0 0 0 1 0 0 0 0 -1 1 1 -1 -1 -1 1 -1 1 1 -1 orthogonal lifted from C2×S5 ρ13 4 -4 4 -4 2 -2 2 -2 0 0 0 0 1 0 0 0 0 -1 -1 1 1 -1 -1 -1 1 1 1 -1 orthogonal lifted from C2×S5 ρ14 4 4 -4 -4 2 2 -2 -2 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 -1 1 1 orthogonal lifted from C2×S5 ρ15 4 4 4 4 2 2 2 2 0 0 0 0 1 0 0 0 0 -1 -1 1 -1 1 1 -1 -1 -1 -1 -1 orthogonal lifted from S5 ρ16 4 -4 -4 4 2 -2 -2 2 0 0 0 0 1 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 -1 1 orthogonal lifted from C2×S5 ρ17 5 -5 5 -5 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 -1 0 0 0 orthogonal lifted from C2×S5 ρ18 5 5 -5 -5 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 0 -1 1 -1 1 -1 1 1 0 0 0 orthogonal lifted from C2×S5 ρ19 5 5 5 5 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 0 -1 -1 -1 -1 -1 -1 -1 0 0 0 orthogonal lifted from S5 ρ20 5 -5 -5 5 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 0 1 1 -1 -1 1 -1 1 0 0 0 orthogonal lifted from C2×S5 ρ21 5 -5 5 -5 -1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 0 -1 -1 1 1 1 -1 1 0 0 0 orthogonal lifted from C2×S5 ρ22 5 5 -5 -5 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 0 1 1 1 1 -1 -1 -1 0 0 0 orthogonal lifted from C2×S5 ρ23 5 5 5 5 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 0 1 -1 1 -1 -1 1 1 0 0 0 orthogonal lifted from S5 ρ24 5 -5 -5 5 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 0 -1 1 1 -1 1 1 -1 0 0 0 orthogonal lifted from C2×S5 ρ25 6 -6 6 -6 0 0 0 0 -2 2 2 -2 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 1 orthogonal lifted from C2×S5 ρ26 6 6 6 6 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 orthogonal lifted from S5 ρ27 6 6 -6 -6 0 0 0 0 -2 -2 2 2 0 0 0 0 0 1 0 0 0 0 0 0 0 1 -1 -1 orthogonal lifted from C2×S5 ρ28 6 -6 -6 6 0 0 0 0 -2 2 -2 2 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 -1 orthogonal lifted from C2×S5

Permutation representations of C22×S5
On 20 points - transitive group 20T117
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 14)(2 11)(3 12)(4 13)(5 20)(6 19)(7 16)(8 17)(9 18)(10 15)
(1 9 5 7)(2 6 4 10)(11 15 13 19)(14 16 20 18)

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

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

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

G:=TransitiveGroup(20,117);

On 24 points - transitive group 24T1345
Generators in S24
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
(1 10)(2 5)(3 18)(4 23)(6 12)(7 11)(8 14)(9 13)(15 19)(16 22)(17 21)(20 24)
(1 4)(2 3)(5 18)(6 15 14 21)(7 22 13 24)(8 19 12 17)(9 16 11 20)(10 23)

G:=sub<Sym(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), (1,10)(2,5)(3,18)(4,23)(6,12)(7,11)(8,14)(9,13)(15,19)(16,22)(17,21)(20,24), (1,4)(2,3)(5,18)(6,15,14,21)(7,22,13,24)(8,19,12,17)(9,16,11,20)(10,23)>;

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), (1,10)(2,5)(3,18)(4,23)(6,12)(7,11)(8,14)(9,13)(15,19)(16,22)(17,21)(20,24), (1,4)(2,3)(5,18)(6,15,14,21)(7,22,13,24)(8,19,12,17)(9,16,11,20)(10,23) );

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)], [(1,10),(2,5),(3,18),(4,23),(6,12),(7,11),(8,14),(9,13),(15,19),(16,22),(17,21),(20,24)], [(1,4),(2,3),(5,18),(6,15,14,21),(7,22,13,24),(8,19,12,17),(9,16,11,20),(10,23)]])

G:=TransitiveGroup(24,1345);

Matrix representation of C22×S5 in GL6(ℤ)

 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,1,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C22×S5 in GAP, Magma, Sage, TeX

C_2^2\times S_5
% in TeX

G:=Group("C2^2xS5");
// GroupNames label

G:=SmallGroup(480,1186);
// by ID

G=gap.SmallGroup(480,1186);
# by ID

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