AGammaL(1,8) - GroupNames

G = AΓL₁(𝔽₈) order 168 = 2³·3·7

Affine semilinear group on 𝔽₈¹

non-abelian, soluble, monomial, A-group

Aliases: AΓL₁(𝔽₈), F₈⋊C₃, C₂³⋊(C₇⋊C₃), Aut(F₈), SmallGroup(168,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — F₈ — AΓL₁(𝔽₈)

Chief series

C₁ — C₂³ — F₈ — AΓL₁(𝔽₈)

Lower central

F₈ — AΓL₁(𝔽₈)

Upper central

C₁

Generators and relations for AΓL₁(𝔽₈)
G = < a,b,c,d,e | a²=b²=c²=d⁷=e³=1, ab=ba, eae^-1=ac=ca, dad^-1=cb=bc, dbd^-1=ebe^-1=a, dcd^-1=b, ece^-1=abc, ede^-1=d⁴ >

C₁

₇C₂

₂₈C₃

₈C₇

₇C₂²

₂₈C₆

₈C₇⋊C₃

C₂³

₇A₄

₇C₂×A₄

F₈

AΓL₁(𝔽₈)

Character table of AΓL₁(𝔽₈)

class	1	2	3A	3B	6A	6B	7A	7B
size	1	7	28	28	28	28	24	24
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	linear of order 3
ρ₃	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	linear of order 3
ρ₄	3	3	0	0	0	0	^-1+√-7/₂	^-1-√-7/₂	complex lifted from C₇⋊C₃
ρ₅	3	3	0	0	0	0	^-1-√-7/₂	^-1+√-7/₂	complex lifted from C₇⋊C₃
ρ₆	7	-1	1	1	-1	-1	0	0	orthogonal faithful
ρ₇	7	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆	0	0	complex faithful
ρ₈	7	-1	ζ₃²	ζ₃	ζ₆	ζ₆⁵	0	0	complex faithful

Permutation representations of AΓL₁(𝔽₈)
►On 8 points: primitive, doubly transitive - transitive group 8T36

Generators in S₈

(1 5)(2 4)(3 6)(7 8)

(1 6)(2 8)(3 5)(4 7)

(1 7)(2 3)(4 6)(5 8)

(2 3 4 5 6 7 8)

(2 7 3)(5 6 8)

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

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

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

G:=TransitiveGroup(8,36);

►On 14 points - transitive group 14T11

Generators in S₁₄

(1 12)(2 13)(5 9)(7 11)

(1 12)(2 13)(3 14)(6 10)

(2 13)(3 14)(4 8)(7 11)

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

(2 3 5)(4 7 6)(8 11 10)(9 13 14)

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

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

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

G:=TransitiveGroup(14,11);

►On 24 points - transitive group 24T283

Generators in S₂₄

(1 12)(2 24)(3 10)(4 8)(5 6)(7 9)(11 16)(13 17)(14 15)(18 22)(19 20)(21 23)

(1 13)(2 18)(3 4)(5 9)(6 7)(8 10)(11 14)(12 17)(15 16)(19 23)(20 21)(22 24)

(1 14)(2 19)(3 5)(4 9)(6 10)(7 8)(11 13)(12 15)(16 17)(18 23)(20 24)(21 22)

(4 5 6 7 8 9 10)(11 12 13 14 15 16 17)(18 19 20 21 22 23 24)

(1 3 2)(4 20 12)(5 22 16)(6 24 13)(7 19 17)(8 21 14)(9 23 11)(10 18 15)

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

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

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

G:=TransitiveGroup(24,283);

►On 28 points - transitive group 28T27

Generators in S₂₈

(1 17)(3 14)(4 20)(5 9)(6 22)(7 23)(8 27)(10 15)(11 16)(12 24)(19 26)(21 28)

(1 24)(2 18)(4 8)(5 21)(6 10)(7 23)(9 28)(11 16)(12 17)(13 25)(15 22)(20 27)

(1 24)(2 25)(3 19)(5 9)(6 15)(7 11)(10 22)(12 17)(13 18)(14 26)(16 23)(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)

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

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

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

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

G:=TransitiveGroup(28,27);

Polynomial with Galois group AΓL₁(𝔽₈) over ℚ

action	f(x)	Disc(f)
8T36	x⁸-28x⁶+12x⁵+194x⁴-80x³-292x²-132x-17	2²⁰·73⁴·1069²
14T11	x¹⁴-27x¹²+171x¹⁰-463x⁸+611x⁶-379x⁴+85x²-1	2²⁶·73⁸

Matrix representation of AΓL₁(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

1	0	0	0	0	0	0
0	0	0	0	1	0	0
0	1	0	0	0	0	0
0	0	0	0	0	1	0
0	0	1	0	0	0	0
0	0	0	0	0	0	1
0	0	0	1	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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

AΓL₁(𝔽₈) in GAP, Magma, Sage, TeX

{\rm AGammaL}_1({\mathbb F}_8)

% in TeX

G:=Group("AGammaL(1,8)");

// GroupNames label

G:=SmallGroup(168,43);

// by ID

G=gap.SmallGroup(168,43);

# by ID

G:=PCGroup([5,-3,-7,-2,2,2,61,1577,217,1263,568,529,884]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=e^3=1,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=c*b=b*c,d*b*d^-1=e*b*e^-1=a,d*c*d^-1=b,e*c*e^-1=a*b*c,e*d*e^-1=d^4>;

// generators/relations

Export

Subgroup lattice of AΓL₁(𝔽₈) in TeX
Character table of AΓL₁(𝔽₈) in TeX

G = AΓL1(𝔽8) order 168 = 23·3·7

Affine semilinear group on 𝔽81

G = AΓL₁(𝔽₈) order 168 = 2³·3·7

Affine semilinear group on 𝔽₈¹