Copied to
clipboard

G = C22×S5order 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: ChiefDerived Lower central Upper central

C1C2C22C22×A5 — C22×S5
A5 — C22×S5
A5 — C22×S5
C1C22

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

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 2)(3 8)(4 9)(5 10)(6 7)(11 16)(12 17)(13 18)(14 15)(19 20)
(1 11 7 13)(2 18 6 16)(3 15 5 19)(4 12)(8 20 10 14)(9 17)

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

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

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

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

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

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

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

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

׿
×
𝔽