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## G = AΓL1(𝔽8)  order 168 = 23·3·7

### Affine semilinear group on 𝔽81

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 >

7C2
28C3
8C7
7C22
28C6
7A4

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
actionf(x)Disc(f)
8T36x8-28x6+12x5+194x4-80x3-292x2-132x-17220·734·10692
14T11x14-27x12+171x10-463x8+611x6-379x4+85x2-1226·738

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

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