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G = C22xS5  order 480 = 25·3·5

Direct product of C22 and S5

direct product, non-abelian, not soluble, rational

Aliases: C22xS5, A5:C23, (C2xA5):C22, (C22xA5):3C2, SmallGroup(480,1186)

Series: Chief►Derived ►Lower central ►Upper central

Chief series
C1 — C2 — C22 — C22xA5 — C22xS5
Derived series
A5 — C22xS5
Lower central
A5 — C22xS5
Upper central
C1 — C22

Subgroups: 2388 in 225 conjugacy classes, 21 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2xC4, D4, C23, D5, C10, A4, D6, C2xC6, C22xC4, C2xD4, C24, F5, D10, C2xC10, S4, C2xA4, C22xS3, C22xC6, C22xD4, C2xF5, C22xD5, C2xS4, C22xA4, S3xC23, A5, C22xF5, C22xS4, S5, C2xA5, C2xS5, C22xA5, C22xS5
Quotients: C1, C2, C22, C23, S5, C2xS5, C22xS5

Character table of C22xS5

 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 C2xS5
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 C2xS5
ρ124-44-4-22-22000010000-111-1-1-11-111-1    orthogonal lifted from C2xS5
ρ134-44-42-22-2000010000-1-111-1-1-1111-1    orthogonal lifted from C2xS5
ρ1444-4-422-2-2000010000-1-1-1-1-1111-111    orthogonal lifted from C2xS5
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 C2xS5
ρ175-55-51-11-11-1-11-1-111-101-1-1111-1000    orthogonal lifted from C2xS5
ρ1855-5-5-1-11111-1-1-1-1-1110-11-11-111000    orthogonal lifted from C2xS5
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 C2xS5
ρ215-55-5-11-111-1-11-11-1-110-1-1111-11000    orthogonal lifted from C2xS5
ρ2255-5-511-1-111-1-1-111-1-101111-1-1-1000    orthogonal lifted from C2xS5
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 C2xS5
ρ256-66-60000-222-20000010000000-1-11    orthogonal lifted from C2xS5
ρ2666660000-2-2-2-20000010000000111    orthogonal lifted from S5
ρ2766-6-60000-2-22200000100000001-1-1    orthogonal lifted from C2xS5
ρ286-6-660000-22-220000010000000-11-1    orthogonal lifted from C2xS5

Permutation representations of C22xS5
►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 C22xS5 ►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] >;
 

C22xS5 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 C22xS5 in TeX

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