Copied to
clipboard

G = F8order 56 = 23·7

Frobenius group

metabelian, soluble, monomial, A-group

Aliases: F8, AGL1(𝔽8), C23⋊C7, SmallGroup(56,11)

Series: Derived Chief Lower central Upper central

C1C23 — F8
C1C23 — F8
C23 — F8
C1

Generators and relations for F8
 G = < a,b,c,d | a2=b2=c2=d7=1, ab=ba, ac=ca, dad-1=cb=bc, dbd-1=a, dcd-1=b >

7C2
8C7
7C22

Character table of F8

 class 127A7B7C7D7E7F
 size 17888888
ρ111111111    trivial
ρ211ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7
ρ311ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7
ρ411ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7
ρ511ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7
ρ611ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7
ρ711ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7
ρ87-1000000    orthogonal faithful

Permutation representations of F8
On 8 points: primitive, sharply doubly transitive - transitive group 8T25
Generators in S8
(1 7)(2 3)(4 6)(5 8)
(1 8)(2 6)(3 4)(5 7)
(1 2)(3 7)(4 5)(6 8)
(2 3 4 5 6 7 8)

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

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

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

G:=TransitiveGroup(8,25);

On 14 points - transitive group 14T6
Generators in S14
(2 13)(4 8)(5 9)(6 10)
(3 14)(5 9)(6 10)(7 11)
(1 12)(4 8)(6 10)(7 11)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

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

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

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

G:=TransitiveGroup(14,6);

On 28 points - transitive group 28T11
Generators in S28
(2 25)(3 10)(4 27)(5 21)(6 15)(7 14)(9 18)(11 20)(12 28)(13 22)(16 23)(19 26)
(1 8)(3 26)(4 11)(5 28)(6 15)(7 16)(10 19)(12 21)(13 22)(14 23)(17 24)(20 27)
(1 17)(2 9)(4 27)(5 12)(6 22)(7 16)(8 24)(11 20)(13 15)(14 23)(18 25)(21 28)
(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)

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

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

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

G:=TransitiveGroup(28,11);

Polynomial with Galois group F8 over ℚ
actionf(x)Disc(f)
8T25x8-2x7-20x6+10x5+102x4+26x3-112x2-50x+7214·174·298
14T6x14+28x12+56x10-245x8-322x6+406x4-56x2-1214·724·318·674·8814

Matrix representation of F8 in GL7(ℤ)

1000000
0100000
00-10000
000-1000
0000-100
0000010
000000-1
,
1000000
0-100000
00-10000
000-1000
0000100
00000-10
0000001
,
-1000000
0-100000
00-10000
0001000
0000-100
0000010
0000001
,
0100000
0010000
0001000
0000100
0000010
0000001
1000000

G:=sub<GL(7,Integers())| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

F8 in GAP, Magma, Sage, TeX

F_8
% in TeX

G:=Group("F8");
// GroupNames label

G:=SmallGroup(56,11);
// by ID

G=gap.SmallGroup(56,11);
# by ID

G:=PCGroup([4,-7,-2,2,2,113,338,563]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^7=1,a*b=b*a,a*c=c*a,d*a*d^-1=c*b=b*c,d*b*d^-1=a,d*c*d^-1=b>;
// generators/relations

׿
×
𝔽