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## G = C24⋊F5order 320 = 26·5

### The semidirect product of C24 and F5 acting faithfully

Aliases: C24⋊F5, C24⋊C5⋊C4, C24⋊D5.C2, SmallGroup(320,1635)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5 — C24⋊F5
 Chief series C1 — C24 — C24⋊C5 — C24⋊D5 — C24⋊F5
 Lower central C24⋊C5 — C24⋊F5
 Upper central C1

Generators and relations for C24⋊F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, fdf-1=ab=ba, ac=ca, ebe-1=fcf-1=ad=da, eae-1=faf-1=d, bc=cb, bd=db, fbf-1=acd, cd=dc, ece-1=abd, ede-1=abcd, fef-1=e3 >

5C2
10C2
20C2
16C5
5C22
10C22
10C22
10C22
10C4
10C22
20C22
20C4
40C4
16D5
5C23
5C23
10C2×C4
10C23
20C2×C4
20C8
20D4
20D4
20D4
16F5
10C2×D4
10C22⋊C4
10C22⋊C4
10M4(2)

Character table of C24⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B size 1 5 10 20 20 40 40 40 64 40 40 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 -i i 1 i -i linear of order 4 ρ4 1 1 1 -1 -1 -1 i -i 1 -i i linear of order 4 ρ5 4 4 4 0 0 0 0 0 -1 0 0 orthogonal lifted from F5 ρ6 5 -3 1 1 1 -1 1 1 0 -1 -1 orthogonal faithful ρ7 5 -3 1 1 1 -1 -1 -1 0 1 1 orthogonal faithful ρ8 5 -3 1 -1 -1 1 -i i 0 -i i complex faithful ρ9 5 -3 1 -1 -1 1 i -i 0 i -i complex faithful ρ10 10 2 -2 2 -2 0 0 0 0 0 0 orthogonal faithful ρ11 10 2 -2 -2 2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C24⋊F5
On 10 points - transitive group 10T24
Generators in S10
```(1 10)(2 6)(4 8)(5 9)
(3 7)(4 8)
(3 7)(5 9)
(1 10)(3 7)(4 8)(5 9)
(1 2 3 4 5)(6 7 8 9 10)
(2 3 5 4)(6 7 9 8)```

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

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

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

`G:=TransitiveGroup(10,24);`

On 10 points - transitive group 10T25
Generators in S10
```(1 8)(2 9)(4 6)(5 7)
(3 10)(4 6)
(3 10)(5 7)
(1 8)(3 10)(4 6)(5 7)
(1 2 3 4 5)(6 7 8 9 10)
(1 8)(2 10 5 6)(3 7 4 9)```

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

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

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

`G:=TransitiveGroup(10,25);`

On 16 points: primitive - transitive group 16T711
Generators in S16
```(1 3)(2 10)(4 11)(5 16)(6 14)(7 9)(8 15)(12 13)
(1 11)(2 8)(3 4)(5 9)(6 13)(7 16)(10 15)(12 14)
(1 16)(2 12)(3 5)(4 9)(6 15)(7 11)(8 14)(10 13)
(1 2)(3 10)(4 15)(5 13)(6 9)(7 14)(8 11)(12 16)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)
(2 3 5 4)(7 15 10 16)(8 12 9 14)(11 13)```

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

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

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

`G:=TransitiveGroup(16,711);`

On 20 points - transitive group 20T77
Generators in S20
```(1 14)(2 16)(4 18)(5 13)(6 20)(7 15)(9 12)(10 19)
(2 7)(3 17)(4 18)(5 10)(8 11)(9 12)(13 19)(15 16)
(1 6)(2 7)(3 11)(5 13)(8 17)(10 19)(14 20)(15 16)
(1 20)(3 17)(4 12)(5 13)(6 14)(8 11)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 20 6 14)(2 17 10 12)(3 19 9 15)(4 16 8 13)(5 18 7 11)```

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

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

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

`G:=TransitiveGroup(20,77);`

On 20 points - transitive group 20T78
Generators in S20
```(2 11)(4 13)(6 18)(7 19)
(2 11)(3 12)(4 13)(5 14)(6 18)(8 20)
(1 15)(2 11)(6 18)(7 19)(8 20)(9 16)
(1 15)(3 12)(6 18)(10 17)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 7 15 19)(2 9 14 17)(3 6 13 20)(4 8 12 18)(5 10 11 16)```

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

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

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

`G:=TransitiveGroup(20,78);`

On 20 points - transitive group 20T79
Generators in S20
```(1 15)(2 11)(4 13)(5 14)(6 20)(8 17)(9 18)(10 19)
(3 12)(4 13)(7 16)(8 17)
(3 12)(5 14)(7 16)(9 18)
(1 15)(3 12)(4 13)(5 14)(7 16)(8 17)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 16 5 17)(3 18 4 20)(6 12 9 13)(7 14 8 11)(10 15)```

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

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

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

`G:=TransitiveGroup(20,79);`

On 20 points - transitive group 20T80
Generators in S20
```(1 14)(2 16)(4 18)(5 13)(6 20)(7 15)(9 12)(10 19)
(2 7)(3 17)(4 18)(5 10)(8 11)(9 12)(13 19)(15 16)
(1 6)(2 7)(3 11)(5 13)(8 17)(10 19)(14 20)(15 16)
(1 20)(3 17)(4 12)(5 13)(6 14)(8 11)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(7 8 10 9)(11 19 12 16)(13 18 15 17)(14 20)```

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

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

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

`G:=TransitiveGroup(20,80);`

On 20 points - transitive group 20T83
Generators in S20
```(2 11)(4 13)(9 18)(10 19)
(2 11)(3 12)(4 13)(5 14)(6 20)(9 18)
(1 15)(2 11)(6 20)(7 16)(9 18)(10 19)
(1 15)(3 12)(8 17)(9 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 16 5 17)(3 18 4 20)(6 12 9 13)(7 14 8 11)(10 15)```

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

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

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

`G:=TransitiveGroup(20,83);`

On 20 points - transitive group 20T88
Generators in S20
```(1 15)(2 11)(4 13)(5 14)(6 20)(8 17)(9 18)(10 19)
(3 12)(4 13)(7 16)(8 17)
(3 12)(5 14)(7 16)(9 18)
(1 15)(3 12)(4 13)(5 14)(7 16)(8 17)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10 15 19)(2 7 14 17)(3 9 13 20)(4 6 12 18)(5 8 11 16)```

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

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

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

`G:=TransitiveGroup(20,88);`

Polynomial with Galois group C24⋊F5 over ℚ
actionf(x)Disc(f)
10T24x10-13x8+56x6-90x4+40x2-4228·536
10T25x10-25x8+160x6-400x4+395x2-125218·513·434

Matrix representation of C24⋊F5 in GL5(ℤ)

 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

`G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;`

C24⋊F5 in GAP, Magma, Sage, TeX

`C_2^4\rtimes F_5`
`% in TeX`

`G:=Group("C2^4:F5");`
`// GroupNames label`

`G:=SmallGroup(320,1635);`
`// by ID`

`G=gap.SmallGroup(320,1635);`
`# by ID`

`G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,170,177,2803,850,1137,9104,1593,5045,4632,2329,986,2463,3695]);`
`// Polycyclic`

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

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