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## G = A4⋊F5order 240 = 24·3·5

### The semidirect product of A4 and F5 acting via F5/D5=C2

Aliases: A4⋊F5, D5.S4, C5⋊(A4⋊C4), C22⋊(C3⋊F5), (C5×A4)⋊1C4, (C2×C10)⋊Dic3, (D5×A4).1C2, (C22×D5).S3, SmallGroup(240,192)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — A4⋊F5
 Chief series C1 — C22 — C2×C10 — C5×A4 — D5×A4 — A4⋊F5
 Lower central C5×A4 — A4⋊F5
 Upper central C1

Generators and relations for A4⋊F5
G = < a,b,c,d,e | a2=b2=c3=d5=e4=1, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

3C2
5C2
15C2
4C3
15C22
15C22
30C4
30C4
20C6
3C10
3D5
4C15
5C23
15C2×C4
15C2×C4
20Dic3
3D10
3D10
6F5
6F5
15C22⋊C4

Character table of A4⋊F5

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 10 15A 15B size 1 3 5 15 8 30 30 30 30 4 40 12 16 16 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 1 -i i i -i 1 -1 1 1 1 linear of order 4 ρ4 1 1 -1 -1 1 i -i -i i 1 -1 1 1 1 linear of order 4 ρ5 2 2 2 2 -1 0 0 0 0 2 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 -2 -1 0 0 0 0 2 1 2 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ7 3 -1 3 -1 0 1 -1 1 -1 3 0 -1 0 0 orthogonal lifted from S4 ρ8 3 -1 3 -1 0 -1 1 -1 1 3 0 -1 0 0 orthogonal lifted from S4 ρ9 3 -1 -3 1 0 -i -i i i 3 0 -1 0 0 complex lifted from A4⋊C4 ρ10 3 -1 -3 1 0 i i -i -i 3 0 -1 0 0 complex lifted from A4⋊C4 ρ11 4 4 0 0 4 0 0 0 0 -1 0 -1 -1 -1 orthogonal lifted from F5 ρ12 4 4 0 0 -2 0 0 0 0 -1 0 -1 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ13 4 4 0 0 -2 0 0 0 0 -1 0 -1 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ14 12 -4 0 0 0 0 0 0 0 -3 0 1 0 0 orthogonal faithful

Permutation representations of A4⋊F5
On 20 points - transitive group 20T61
Generators in S20
```(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 10)(2 6)(3 7)(4 8)(5 9)(11 16)(12 17)(13 18)(14 19)(15 20)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(6 7 9 8)(11 17 14 18)(12 19 13 16)(15 20)```

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

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

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

`G:=TransitiveGroup(20,61);`

On 30 points - transitive group 30T53
Generators in S30
```(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(2 3 5 4)(6 24 10 22)(7 21 9 25)(8 23)(11 14 15 12)(16 29 20 27)(17 26 19 30)(18 28)```

`G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,29,20,27)(17,26,19,30)(18,28)>;`

`G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,29,20,27)(17,26,19,30)(18,28) );`

`G=PermutationGroup([(1,13),(2,14),(3,15),(4,11),(5,12),(6,26),(7,27),(8,28),(9,29),(10,30)], [(6,26),(7,27),(8,28),(9,29),(10,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,21,11),(7,22,12),(8,23,13),(9,24,14),(10,25,15)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(2,3,5,4),(6,24,10,22),(7,21,9,25),(8,23),(11,14,15,12),(16,29,20,27),(17,26,19,30),(18,28)])`

`G:=TransitiveGroup(30,53);`

On 30 points - transitive group 30T64
Generators in S30
```(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 13)(2 15 5 11)(3 12 4 14)(6 19 10 17)(7 16 9 20)(8 18)(21 29 25 27)(22 26 24 30)(23 28)```

`G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,13)(2,15,5,11)(3,12,4,14)(6,19,10,17)(7,16,9,20)(8,18)(21,29,25,27)(22,26,24,30)(23,28)>;`

`G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,13)(2,15,5,11)(3,12,4,14)(6,19,10,17)(7,16,9,20)(8,18)(21,29,25,27)(22,26,24,30)(23,28) );`

`G=PermutationGroup([(1,13),(2,14),(3,15),(4,11),(5,12),(6,26),(7,27),(8,28),(9,29),(10,30)], [(6,26),(7,27),(8,28),(9,29),(10,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,21,11),(7,22,12),(8,23,13),(9,24,14),(10,25,15)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,13),(2,15,5,11),(3,12,4,14),(6,19,10,17),(7,16,9,20),(8,18),(21,29,25,27),(22,26,24,30),(23,28)])`

`G:=TransitiveGroup(30,64);`

A4⋊F5 is a maximal subgroup of   F5×S4
A4⋊F5 is a maximal quotient of   C5⋊U2(𝔽3)  D10.S4  Dic5.S4

Matrix representation of A4⋊F5 in GL7(𝔽61)

 60 0 0 0 0 0 0 60 0 1 0 0 0 0 60 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 1 60 0 0 0 0 1 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 60 0 0 0 1 0 0 60 0 0 0 0 1 0 60 0 0 0 0 0 1 60
,
 0 11 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

`G:=sub<GL(7,GF(61))| [60,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60],[0,11,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;`

A4⋊F5 in GAP, Magma, Sage, TeX

`A_4\rtimes F_5`
`% in TeX`

`G:=Group("A4:F5");`
`// GroupNames label`

`G:=SmallGroup(240,192);`
`// by ID`

`G=gap.SmallGroup(240,192);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-5,-2,2,12,146,867,585,3604,916,2165,1637]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^5=e^4=1,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;`
`// generators/relations`

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