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G = F9order 72 = 23·32

Frobenius group

Aliases: F9, AGL1(𝔽9), C32⋊C8, C3⋊S3.C4, C32⋊C4.1C2, SmallGroup(72,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — F9
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — F9
 Lower central C32 — F9
 Upper central C1

Generators and relations for F9
G = < a,b,c | a3=b3=c8=1, cac-1=ab=ba, cbc-1=a >

Character table of F9

 class 1 2 3 4A 4B 8A 8B 8C 8D size 1 9 8 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -i -i i i linear of order 4 ρ4 1 1 1 -1 -1 i i -i -i linear of order 4 ρ5 1 -1 1 -i i ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ6 1 -1 1 i -i ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ7 1 -1 1 i -i ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ8 1 -1 1 -i i ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ9 8 0 -1 0 0 0 0 0 0 orthogonal faithful

Permutation representations of F9
On 9 points: primitive, sharply doubly transitive - transitive group 9T15
Generators in S9
```(1 5 9)(2 7 8)(3 6 4)
(1 6 2)(3 8 9)(4 7 5)
(2 3 4 5 6 7 8 9)```

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

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

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

`G:=TransitiveGroup(9,15);`

On 12 points - transitive group 12T46
Generators in S12
```(1 12 8)(3 6 10)(4 7 11)
(1 8 12)(2 5 9)(4 7 11)
(1 2 3 4)(5 6 7 8 9 10 11 12)```

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

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

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

`G:=TransitiveGroup(12,46);`

On 18 points - transitive group 18T28
Generators in S18
```(1 16 12)(2 10 6)(3 9 14)(4 11 13)(5 7 18)(8 17 15)
(1 3 7)(2 17 13)(4 10 15)(5 12 14)(6 8 11)(9 18 16)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)```

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

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

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

`G:=TransitiveGroup(18,28);`

On 24 points - transitive group 24T81
Generators in S24
```(2 22 10)(3 23 11)(4 12 24)(6 14 18)(7 15 19)(8 20 16)
(1 21 9)(3 23 11)(4 24 12)(5 13 17)(7 15 19)(8 16 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)```

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

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

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

`G:=TransitiveGroup(24,81);`

F9 is a maximal subgroup of   AΓL1(𝔽9)  C3⋊F9  C52F9  C5⋊F9
F9 is a maximal quotient of   C2.F9  He3⋊C8  C3⋊F9  C52F9  C5⋊F9

Polynomial with Galois group F9 over ℚ
actionf(x)Disc(f)
9T15x9-72x7+1464x5-960x4-8928x3+13440x2-2064x-2560267·312·56·72·2392·5032
12T46x12+12x10-19x9+54x8-171x7+169x6-513x5+447x4-573x3+549x2-180x+16216·312·138·1710

Matrix representation of F9 in GL8(ℤ)

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

`G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0],[0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,1,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,1,0,-1,0] >;`

F9 in GAP, Magma, Sage, TeX

`F_9`
`% in TeX`

`G:=Group("F9");`
`// GroupNames label`

`G:=SmallGroup(72,39);`
`// by ID`

`G=gap.SmallGroup(72,39);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,3,10,26,483,568,93,1404,809,314]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^3=c^8=1,c*a*c^-1=a*b=b*a,c*b*c^-1=a>;`
`// generators/relations`

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