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G = A5⋊C4order 240 = 24·3·5

The semidirect product of A5 and C4 acting via C4/C2=C2

non-abelian, not soluble

Aliases: A5⋊C4, C2.1S5, (C2×A5).C2, SmallGroup(240,91)

Series: ChiefDerived Lower central Upper central

C1C2C2×A5 — A5⋊C4
A5 — A5⋊C4
A5 — A5⋊C4
C1C2

15C2
15C2
10C3
6C5
5C22
10C4
15C22
15C22
30C4
10C6
10S3
10S3
6D5
6C10
6D5
5C23
15C2×C4
15C2×C4
5A4
10Dic3
10D6
10C12
6F5
6D10
6F5
15C22⋊C4
5C2×A4
10C4×S3
6C2×F5
5A4⋊C4

Character table of A5⋊C4

 class 12A2B2C34A4B4C4D561012A12B
 size 11151520101030302420242020
ρ111111111111111    trivial
ρ211111-1-1-1-1111-1-1    linear of order 2
ρ31-1-111i-i-ii1-1-1-ii    linear of order 4
ρ41-1-111-iii-i1-1-1i-i    linear of order 4
ρ5440012200-11-1-1-1    orthogonal lifted from S5
ρ644001-2-200-11-111    orthogonal lifted from S5
ρ74-4001-2i2i00-1-11-ii    complex faithful
ρ84-40012i-2i00-1-11i-i    complex faithful
ρ95511-1-1-1110-10-1-1    orthogonal lifted from S5
ρ105511-111-1-10-1011    orthogonal lifted from S5
ρ115-5-11-1i-ii-i010-ii    complex faithful
ρ125-5-11-1-ii-ii010i-i    complex faithful
ρ136-62-20000010-100    orthogonal faithful
ρ1466-2-20000010100    orthogonal lifted from S5

Permutation representations of A5⋊C4
On 12 points - transitive group 12T124
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 11 2 3)(4 9 12 6)(5 7 8 10)

G:=sub<Sym(12)| (3,4,5,6,7)(8,9,10,11,12), (1,11,2,3)(4,9,12,6)(5,7,8,10)>;

G:=Group( (3,4,5,6,7)(8,9,10,11,12), (1,11,2,3)(4,9,12,6)(5,7,8,10) );

G=PermutationGroup([(3,4,5,6,7),(8,9,10,11,12)], [(1,11,2,3),(4,9,12,6),(5,7,8,10)])

G:=TransitiveGroup(12,124);

On 20 points - transitive group 20T66
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19 11 10)(2 18 12 9)(3 16 13 7)(4 17 14 8)(5 20 15 6)

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

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

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

G:=TransitiveGroup(20,66);

On 24 points - transitive group 24T571
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 24 3 8)(2 19 4 10)(5 18 21 14)(6 15 22 11)(7 16 23 12)(9 17 20 13)

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

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

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

G:=TransitiveGroup(24,571);

On 24 points - transitive group 24T578
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 24 4 14)(2 18 3 8)(5 23 15 13)(6 20 16 10)(7 11 17 21)(9 12 19 22)

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

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

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

G:=TransitiveGroup(24,578);

A5⋊C4 is a maximal subgroup of   C4×S5  A5⋊Q8  C22⋊S5
A5⋊C4 is a maximal quotient of   A5⋊C8  GL2(𝔽5)  C22.2S5

Polynomial with Galois group A5⋊C4 over ℚ
actionf(x)Disc(f)
12T124x12-6x11+14x10-15x9+5x8+4x7-4x6+x5-4x2+4x-1515·134

Matrix representation of A5⋊C4 in GL3(𝔽5) generated by

031
113
022
,
312
132
221
G:=sub<GL(3,GF(5))| [0,1,0,3,1,2,1,3,2],[3,1,2,1,3,2,2,2,1] >;

A5⋊C4 in GAP, Magma, Sage, TeX

A_5\rtimes C_4
% in TeX

G:=Group("A5:C4");
// GroupNames label

G:=SmallGroup(240,91);
// by ID

G=gap.SmallGroup(240,91);
# by ID

Export

Subgroup lattice of A5⋊C4 in TeX
Character table of A5⋊C4 in TeX

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