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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

C1C23F8 — AΓL1(𝔽8)
C1C23F8 — AΓL1(𝔽8)
F8 — AΓL1(𝔽8)
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
8C7⋊C3
7A4
7C2×A4

Character table of AΓL1(𝔽8)

 class 123A3B6A6B7A7B
 size 17282828282424
ρ111111111    trivial
ρ211ζ3ζ32ζ3ζ3211    linear of order 3
ρ311ζ32ζ3ζ32ζ311    linear of order 3
ρ4330000-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ5330000-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ67-111-1-100    orthogonal faithful
ρ77-1ζ3ζ32ζ65ζ600    complex faithful
ρ87-1ζ32ζ3ζ6ζ6500    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 23)(3 10)(4 8)(5 6)(7 9)(11 16)(13 17)(14 15)(18 19)(20 22)(21 24)
(1 13)(2 24)(3 4)(5 9)(6 7)(8 10)(11 14)(12 17)(15 16)(18 22)(19 20)(21 23)
(1 14)(2 18)(3 5)(4 9)(6 10)(7 8)(11 13)(12 15)(16 17)(19 23)(20 21)(22 24)
(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 19 12)(5 21 16)(6 23 13)(7 18 17)(8 20 14)(9 22 11)(10 24 15)

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

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

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

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)

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

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

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