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## G = F16order 240 = 24·3·5

### Frobenius group

Aliases: F16, AGL1(𝔽16), C24⋊C15, C24⋊C5⋊C3, C22⋊A4⋊C5, SmallGroup(240,191)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — F16
 Chief series C1 — C24 — C24⋊C5 — F16
 Lower central C24 — F16
 Upper central C1

Generators and relations for F16
G = < a,b,c,d,e | a2=b2=c2=d2=e15=1, ab=ba, ac=ca, ad=da, eae-1=bcd, ebe-1=bc=cb, ede-1=bd=db, cd=dc, ece-1=a >

15C2
16C3
16C5
5C22
15C22
15C22
16C15
15C23
20A4

Character table of F16

 class 1 2 3A 3B 5A 5B 5C 5D 15A 15B 15C 15D 15E 15F 15G 15H size 1 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ3 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 ζ54 ζ52 ζ53 ζ5 ζ52 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ5 linear of order 5 ρ5 1 1 1 1 ζ53 ζ54 ζ5 ζ52 ζ54 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ52 linear of order 5 ρ6 1 1 1 1 ζ52 ζ5 ζ54 ζ53 ζ5 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ53 linear of order 5 ρ7 1 1 1 1 ζ5 ζ53 ζ52 ζ54 ζ53 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ54 linear of order 5 ρ8 1 1 ζ3 ζ32 ζ5 ζ53 ζ52 ζ54 ζ3ζ53 ζ32ζ5 ζ32ζ53 ζ32ζ52 ζ3ζ52 ζ3ζ54 ζ3ζ5 ζ32ζ54 linear of order 15 ρ9 1 1 ζ32 ζ3 ζ52 ζ5 ζ54 ζ53 ζ32ζ5 ζ3ζ52 ζ3ζ5 ζ3ζ54 ζ32ζ54 ζ32ζ53 ζ32ζ52 ζ3ζ53 linear of order 15 ρ10 1 1 ζ3 ζ32 ζ52 ζ5 ζ54 ζ53 ζ3ζ5 ζ32ζ52 ζ32ζ5 ζ32ζ54 ζ3ζ54 ζ3ζ53 ζ3ζ52 ζ32ζ53 linear of order 15 ρ11 1 1 ζ32 ζ3 ζ53 ζ54 ζ5 ζ52 ζ32ζ54 ζ3ζ53 ζ3ζ54 ζ3ζ5 ζ32ζ5 ζ32ζ52 ζ32ζ53 ζ3ζ52 linear of order 15 ρ12 1 1 ζ3 ζ32 ζ53 ζ54 ζ5 ζ52 ζ3ζ54 ζ32ζ53 ζ32ζ54 ζ32ζ5 ζ3ζ5 ζ3ζ52 ζ3ζ53 ζ32ζ52 linear of order 15 ρ13 1 1 ζ32 ζ3 ζ5 ζ53 ζ52 ζ54 ζ32ζ53 ζ3ζ5 ζ3ζ53 ζ3ζ52 ζ32ζ52 ζ32ζ54 ζ32ζ5 ζ3ζ54 linear of order 15 ρ14 1 1 ζ3 ζ32 ζ54 ζ52 ζ53 ζ5 ζ3ζ52 ζ32ζ54 ζ32ζ52 ζ32ζ53 ζ3ζ53 ζ3ζ5 ζ3ζ54 ζ32ζ5 linear of order 15 ρ15 1 1 ζ32 ζ3 ζ54 ζ52 ζ53 ζ5 ζ32ζ52 ζ3ζ54 ζ3ζ52 ζ3ζ53 ζ32ζ53 ζ32ζ5 ζ32ζ54 ζ3ζ5 linear of order 15 ρ16 15 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of F16
On 16 points: primitive, sharply doubly transitive - transitive group 16T447
Generators in S16
```(1 14)(2 3)(4 9)(5 7)(6 12)(8 16)(10 13)(11 15)
(1 4)(2 11)(3 15)(5 16)(6 13)(7 8)(9 14)(10 12)
(1 15)(2 9)(3 4)(5 10)(6 8)(7 13)(11 14)(12 16)
(1 8)(2 10)(3 13)(4 7)(5 9)(6 15)(11 12)(14 16)
(2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,447);`

On 20 points - transitive group 20T67
Generators in S20
```(1 10)(3 7)(4 8)(5 14)(9 19)(12 17)(13 18)(15 20)
(1 15)(3 12)(4 13)(5 19)(7 17)(8 18)(9 14)(10 20)
(1 15)(2 11)(4 8)(5 9)(6 16)(10 20)(13 18)(14 19)
(2 16)(3 17)(4 8)(5 19)(6 11)(7 12)(9 14)(13 18)
(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)```

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

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

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

`G:=TransitiveGroup(20,67);`

On 30 points - transitive group 30T50
Generators in S30
```(3 23)(6 26)(7 27)(9 29)(11 16)(12 17)(13 18)(14 19)
(1 21)(2 22)(3 23)(4 24)(8 28)(11 16)(12 17)(14 19)
(4 24)(7 27)(8 28)(10 30)(12 17)(13 18)(14 19)(15 20)
(1 21)(3 23)(5 25)(6 26)(7 27)(8 28)(12 17)(15 20)
(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)```

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

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

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

`G:=TransitiveGroup(30,50);`

F16 is a maximal subgroup of   F16⋊C2

Matrix representation of F16 in GL15(ℤ)

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

`G:=sub<GL(15,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0],[0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0] >;`

F16 in GAP, Magma, Sage, TeX

`F_{16}`
`% in TeX`

`G:=Group("F16");`
`// GroupNames label`

`G:=SmallGroup(240,191);`
`// by ID`

`G=gap.SmallGroup(240,191);`
`# by ID`

`G:=PCGroup([6,-3,-5,-2,2,2,2,2972,1358,5403,849,3154,2110,7565,911]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^15=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b*c*d,e*b*e^-1=b*c=c*b,e*d*e^-1=b*d=d*b,c*d=d*c,e*c*e^-1=a>;`
`// generators/relations`

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