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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

Subgroups: 2388 in 225 conjugacy classes, 21 normal (5 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22, C22 [×27], C5, S3 [×8], C6 [×7], C2×C4 [×6], D4 [×16], C23 [×17], D5 [×4], C10 [×3], A4, D6 [×28], C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], F5 [×4], D10 [×6], C2×C10, S4 [×4], C2×A4 [×3], C22×S3 [×14], C22×C6, C22×D4, C2×F5 [×6], C22×D5, C2×S4 [×6], C22×A4, S3×C23, A5, C22×F5, C22×S4, S5 [×4], C2×A5 [×3], C2×S5 [×6], C22×A5, C22×S5

Quotients:
C1, C2 [×7], C22 [×7], C23, S5, C2×S5 [×3], C22×S5

Permutation representations
►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 12)(2 19)(3 20)(4 11)(5 18)(6 17)(7 14)(8 15)(9 16)(10 13)

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

(5 9 7 13)(8 10 14 12)(15 17 21 19)(16 24 20 22)
 
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), (1,11)(2,6)(3,18)(4,23)(5,16)(7,20)(8,19)(9,22)(10,21)(12,15)(13,24)(14,17), (5,9,7,13)(8,10,14,12)(15,17,21,19)(16,24,20,22)>;
 
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), (1,11)(2,6)(3,18)(4,23)(5,16)(7,20)(8,19)(9,22)(10,21)(12,15)(13,24)(14,17), (5,9,7,13)(8,10,14,12)(15,17,21,19)(16,24,20,22) );
 
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)], [(1,11),(2,6),(3,18),(4,23),(5,16),(7,20),(8,19),(9,22),(10,21),(12,15),(13,24),(14,17)], [(5,9,7,13),(8,10,14,12),(15,17,21,19),(16,24,20,22)])
 
G:=TransitiveGroup(24,1345);
 

Matrix representation ►G ⊆ 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] >;
 

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

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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