metabelian, soluble, monomial, A-group
Aliases: F9, AGL1(𝔽9), C32⋊C8, C3⋊S3.C4, C32⋊C4.1C2, SmallGroup(72,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — F9 |
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 |
►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 C5⋊2F9 C5⋊F9
F9 is a maximal quotient of C2.F9 He3⋊C8 C3⋊F9 C5⋊2F9 C5⋊F9
| action | f(x) | Disc(f) |
|---|---|---|
| 9T15 | x9-72x7+1464x5-960x4-8928x3+13440x2-2064x-2560 | 267·312·56·72·2392·5032 |
| 12T46 | x12+12x10-19x9+54x8-171x7+169x6-513x5+447x4-573x3+549x2-180x+16 | 216·312·138·1710 |
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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