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G = C22×S5  order 480 = 25·3·5

Direct product of C22 and S5

direct product, non-abelian, not soluble, rational

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

Series: Chief►Derived ►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 12A2B2C2D2E2F2G2H2I2J2K34A4B4C4D56A6B6C6D6E6F6G10A10B10C
 size 1111101010101515151520303030302420202020202020242424
ρ11111111111111111111111111111    trivial
ρ211-1-1-1-11111-1-1111-1-11-1-1-1-11111-1-1    linear of order 2
ρ31-1-11-111-11-11-111-11-11-1-111-11-1-11-1    linear of order 2
ρ41-11-11-11-11-1-1111-1-11111-1-1-11-1-1-11    linear of order 2
ρ511-1-111-1-111-1-11-1-11111-11-11-1-11-1-1    linear of order 2
ρ61111-1-1-1-111111-1-1-1-11-11-111-1-1111    linear of order 2
ρ71-1-111-1-111-11-11-11-1111-1-11-1-11-11-1    linear of order 2
ρ81-11-1-11-111-1-111-111-11-111-1-1-11-1-11    linear of order 2
ρ94-4-44-222-2000010000-11-1-11-1-111-11    orthogonal lifted from C2×S5
ρ104444-2-2-2-2000010000-11111111-1-1-1    orthogonal lifted from S5
ρ1144-4-4-2-222000010000-11-11-11-1-1-111    orthogonal lifted from C2×S5
ρ124-44-4-22-22000010000-111-1-1-11-111-1    orthogonal lifted from C2×S5
ρ134-44-42-22-2000010000-1-111-1-1-1111-1    orthogonal lifted from C2×S5
ρ1444-4-422-2-2000010000-1-1-1-1-1111-111    orthogonal lifted from C2×S5
ρ1544442222000010000-1-11-111-1-1-1-1-1    orthogonal lifted from S5
ρ164-4-442-2-22000010000-1-1-111-11-11-11    orthogonal lifted from C2×S5
ρ175-55-51-11-11-1-11-1-111-101-1-1111-1000    orthogonal lifted from C2×S5
ρ1855-5-5-1-11111-1-1-1-1-1110-11-11-111000    orthogonal lifted from C2×S5
ρ195555-1-1-1-11111-111110-1-1-1-1-1-1-1000    orthogonal lifted from S5
ρ205-5-551-1-111-11-1-11-11-1011-1-11-11000    orthogonal lifted from C2×S5
ρ215-55-5-11-111-1-11-11-1-110-1-1111-11000    orthogonal lifted from C2×S5
ρ2255-5-511-1-111-1-1-111-1-101111-1-1-1000    orthogonal lifted from C2×S5
ρ23555511111111-1-1-1-1-101-11-1-111000    orthogonal lifted from S5
ρ245-5-55-111-11-11-1-1-11-110-111-111-1000    orthogonal lifted from C2×S5
ρ256-66-60000-222-20000010000000-1-11    orthogonal lifted from C2×S5
ρ2666660000-2-2-2-20000010000000111    orthogonal lifted from S5
ρ2766-6-60000-2-22200000100000001-1-1    orthogonal lifted from C2×S5
ρ286-6-660000-22-220000010000000-11-1    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)

-100000
010000
000010
001000
000001
00-1-1-1-1
,
100000
0-10000
000010
000100
001000
000001
,
100000
010000
000100
000010
000001
001000

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
 

Export

Character table of C22×S5 in TeX

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