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## G = C22×A5order 240 = 24·3·5

### Direct product of C22 and A5

Aliases: C22×A5, SmallGroup(240,190)

Series: ChiefDerived Lower central Upper central

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

Subgroups: 601 in 71 conjugacy classes, 10 normal (4 characteristic)
C1, C2 [×3], C2 [×4], C3, C22, C22 [×12], C5, S3 [×4], C6 [×3], C23 [×7], D5 [×4], C10 [×3], A4, D6 [×6], C2×C6, C24, D10 [×6], C2×C10, C2×A4 [×3], C22×S3, C22×D5, C22×A4, A5, C2×A5 [×3], C22×A5
Quotients: C1, C2 [×3], C22, A5, C2×A5 [×3], C22×A5

Character table of C22×A5

 class 1 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F size 1 1 1 1 15 15 15 15 20 12 12 20 20 20 12 12 12 12 12 12 ρ1 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 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 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 linear of order 2 ρ5 3 -3 3 -3 1 -1 -1 1 0 1+√5/2 1-√5/2 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 orthogonal lifted from C2×A5 ρ6 3 3 3 3 -1 -1 -1 -1 0 1-√5/2 1+√5/2 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from A5 ρ7 3 -3 -3 3 -1 -1 1 1 0 1+√5/2 1-√5/2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from C2×A5 ρ8 3 -3 3 -3 1 -1 -1 1 0 1-√5/2 1+√5/2 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 orthogonal lifted from C2×A5 ρ9 3 3 -3 -3 1 -1 1 -1 0 1+√5/2 1-√5/2 0 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from C2×A5 ρ10 3 3 3 3 -1 -1 -1 -1 0 1+√5/2 1-√5/2 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from A5 ρ11 3 3 -3 -3 1 -1 1 -1 0 1-√5/2 1+√5/2 0 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from C2×A5 ρ12 3 -3 -3 3 -1 -1 1 1 0 1-√5/2 1+√5/2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from C2×A5 ρ13 4 4 -4 -4 0 0 0 0 1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×A5 ρ14 4 -4 4 -4 0 0 0 0 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from C2×A5 ρ15 4 4 4 4 0 0 0 0 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A5 ρ16 4 -4 -4 4 0 0 0 0 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 orthogonal lifted from C2×A5 ρ17 5 -5 -5 5 1 1 -1 -1 -1 0 0 1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A5 ρ18 5 5 5 5 1 1 1 1 -1 0 0 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A5 ρ19 5 5 -5 -5 -1 1 -1 1 -1 0 0 1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A5 ρ20 5 -5 5 -5 -1 1 1 -1 -1 0 0 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A5

Permutation representations of C22×A5
On 20 points - transitive group 20T64
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 18)(4 11)(5 20)(6 17)(7 14)(8 13)(9 16)(10 15)

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)(3,18)(4,11)(5,20)(6,17)(7,14)(8,13)(9,16)(10,15)>;

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)(3,18)(4,11)(5,20)(6,17)(7,14)(8,13)(9,16)(10,15) );

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),(3,18),(4,11),(5,20),(6,17),(7,14),(8,13),(9,16),(10,15)])

G:=TransitiveGroup(20,64);

On 24 points - transitive group 24T572
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 24)(2 19)(3 9)(4 14)(5 18)(8 15)(10 23)(13 20)

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), (1,24)(2,19)(3,9)(4,14)(5,18)(8,15)(10,23)(13,20)>;

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), (1,24)(2,19)(3,9)(4,14)(5,18)(8,15)(10,23)(13,20) );

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)], [(1,24),(2,19),(3,9),(4,14),(5,18),(8,15),(10,23),(13,20)])

G:=TransitiveGroup(24,572);

On 24 points - transitive group 24T573
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 8)(2 13)(3 15)(4 20)(5 22)(6 23)(7 16)(9 24)(10 17)(11 18)(12 21)(14 19)

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

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

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

G:=TransitiveGroup(24,573);

C22×A5 is a maximal subgroup of   C22⋊S5
C22×A5 is a maximal quotient of   D4.A5  Q8.A5

Matrix representation of C22×A5 in GL5(𝔽31)

 1 0 0 0 0 0 30 0 0 0 0 0 29 3 10 0 0 28 21 29 0 0 1 0 0
,
 30 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 30

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,30,0,0,0,0,0,29,28,1,0,0,3,21,0,0,0,10,29,0],[30,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,30] >;

C22×A5 in GAP, Magma, Sage, TeX

C_2^2\times A_5
% in TeX

G:=Group("C2^2xA5");
// GroupNames label

G:=SmallGroup(240,190);
// by ID

G=gap.SmallGroup(240,190);
# by ID

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