metabelian, soluble, monomial, A-group
Aliases: F8, AGL1(𝔽8), C23⋊C7, SmallGroup(56,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — F8 |
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 >
Character table of F8
| class | 1 | 2 | 7A | 7B | 7C | 7D | 7E | 7F | |
| size | 1 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | ζ74 | ζ76 | ζ72 | ζ75 | ζ7 | ζ73 | linear of order 7 |
| ρ3 | 1 | 1 | ζ72 | ζ73 | ζ7 | ζ76 | ζ74 | ζ75 | linear of order 7 |
| ρ4 | 1 | 1 | ζ75 | ζ74 | ζ76 | ζ7 | ζ73 | ζ72 | linear of order 7 |
| ρ5 | 1 | 1 | ζ73 | ζ7 | ζ75 | ζ72 | ζ76 | ζ74 | linear of order 7 |
| ρ6 | 1 | 1 | ζ7 | ζ75 | ζ74 | ζ73 | ζ72 | ζ76 | linear of order 7 |
| ρ7 | 1 | 1 | ζ76 | ζ72 | ζ73 | ζ74 | ζ75 | ζ7 | linear of order 7 |
| ρ8 | 7 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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 17)(6 18)(7 14)(9 21)(11 16)(12 28)(13 22)(15 26)(19 23)
(1 8)(3 26)(4 11)(5 28)(6 18)(7 19)(10 15)(12 17)(13 22)(14 23)(16 27)(20 24)
(1 20)(2 9)(4 27)(5 12)(6 22)(7 19)(8 24)(11 16)(13 18)(14 23)(17 28)(21 25)
(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)(5,17)(6,18)(7,14)(9,21)(11,16)(12,28)(13,22)(15,26)(19,23), (1,8)(3,26)(4,11)(5,28)(6,18)(7,19)(10,15)(12,17)(13,22)(14,23)(16,27)(20,24), (1,20)(2,9)(4,27)(5,12)(6,22)(7,19)(8,24)(11,16)(13,18)(14,23)(17,28)(21,25), (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)(5,17)(6,18)(7,14)(9,21)(11,16)(12,28)(13,22)(15,26)(19,23), (1,8)(3,26)(4,11)(5,28)(6,18)(7,19)(10,15)(12,17)(13,22)(14,23)(16,27)(20,24), (1,20)(2,9)(4,27)(5,12)(6,22)(7,19)(8,24)(11,16)(13,18)(14,23)(17,28)(21,25), (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),(5,17),(6,18),(7,14),(9,21),(11,16),(12,28),(13,22),(15,26),(19,23)], [(1,8),(3,26),(4,11),(5,28),(6,18),(7,19),(10,15),(12,17),(13,22),(14,23),(16,27),(20,24)], [(1,20),(2,9),(4,27),(5,12),(6,22),(7,19),(8,24),(11,16),(13,18),(14,23),(17,28),(21,25)], [(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);
F8 is a maximal subgroup of
AΓL1(𝔽8) C43⋊C7 C23.F8 C26⋊C7 C23⋊F8
F8 is a maximal quotient of C7.F8 C43⋊C7 C23.F8 C26⋊C7 C23⋊F8
| action | f(x) | Disc(f) |
|---|---|---|
| 8T25 | x8-2x7-20x6+10x5+102x4+26x3-112x2-50x+7 | 214·174·298 |
| 14T6 | x14+28x12+56x10-245x8-322x6+406x4-56x2-1 | 214·724·318·674·8814 |
Matrix representation of F8 ►in GL7(ℤ)
| 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 | 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 |
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
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