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G = F9order 72 = 23·32

Frobenius group

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

C1C32 — F9
C1C32C3⋊S3C32⋊C4 — F9
C32 — F9
C1

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

9C2
4C3
9C4
12S3
9C8

Character table of F9

 class 1234A4B8A8B8C8D
 size 198999999
ρ1111111111    trivial
ρ211111-1-1-1-1    linear of order 2
ρ3111-1-1-i-iii    linear of order 4
ρ4111-1-1ii-i-i    linear of order 4
ρ51-11-iiζ85ζ8ζ87ζ83    linear of order 8
ρ61-11i-iζ87ζ83ζ85ζ8    linear of order 8
ρ71-11i-iζ83ζ87ζ8ζ85    linear of order 8
ρ81-11-iiζ8ζ85ζ83ζ87    linear of order 8
ρ980-1000000    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 8 12)(2 5 9)(4 7 11)
(1 8 12)(2 9 5)(3 6 10)
(1 2 3 4)(5 6 7 8 9 10 11 12)

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

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

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

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

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

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

00010000
00001000
00000100
00000010
00000001
-1-1-1-1-1-1-1-1
10000000
01000000
,
01000000
00100000
10000000
00001000
00000100
00010000
00000001
-1-1-1-1-1-1-1-1
,
10000000
00010000
00000010
00001000
00000001
01000000
-1-1-1-1-1-1-1-1
00100000

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

Export

Subgroup lattice of F9 in TeX
Character table of F9 in TeX

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