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## G = C2×AΓL1(𝔽8)  order 336 = 24·3·7

### Direct product of C2 and AΓL1(𝔽8)

Aliases: C2×AΓL1(𝔽8), F8⋊C6, (C2×F8)⋊C3, C24⋊(C7⋊C3), C23⋊(C2×C7⋊C3), SmallGroup(336,210)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — F8 — C2×AΓL1(𝔽8)
 Chief series C1 — C23 — F8 — AΓL1(𝔽8) — C2×AΓL1(𝔽8)
 Lower central F8 — C2×AΓL1(𝔽8)
 Upper central C1 — C2

Generators and relations for C2×AΓL1(𝔽8)
G = < a,b,c,d,e,f | a2=b2=c2=d2=e7=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, ebe-1=dc=cd, ece-1=fcf-1=b, ede-1=c, fdf-1=bcd, fef-1=e4 >

Character table of C2×AΓL1(𝔽8)

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 7A 7B 14A 14B size 1 1 7 7 28 28 28 28 28 28 28 28 24 24 24 24 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ5 1 -1 -1 1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 1 1 -1 -1 linear of order 6 ρ6 1 -1 -1 1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 1 1 -1 -1 linear of order 6 ρ7 3 3 3 3 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ8 3 -3 -3 3 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ9 3 -3 -3 3 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ10 3 3 3 3 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ11 7 7 -1 -1 1 1 -1 -1 -1 1 -1 1 0 0 0 0 orthogonal lifted from AΓL1(𝔽8) ρ12 7 -7 1 -1 1 1 1 -1 -1 -1 1 -1 0 0 0 0 orthogonal faithful ρ13 7 7 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ3 ζ65 ζ32 0 0 0 0 complex lifted from AΓL1(𝔽8) ρ14 7 -7 1 -1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 0 0 0 0 complex faithful ρ15 7 -7 1 -1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 0 0 0 0 complex faithful ρ16 7 7 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ32 ζ6 ζ3 0 0 0 0 complex lifted from AΓL1(𝔽8)

Permutation representations of C2×AΓL1(𝔽8)
On 14 points - transitive group 14T18
Generators in S14
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 8)
(1 9)(2 10)(5 13)(7 8)
(1 9)(2 10)(3 11)(6 14)
(2 10)(3 11)(4 12)(7 8)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(2 3 5)(4 7 6)(8 14 12)(10 11 13)

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

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

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

G:=TransitiveGroup(14,18);

On 16 points - transitive group 16T712
Generators in S16
(1 2)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)
(1 3)(2 10)(4 8)(5 6)(7 9)(11 15)(12 13)(14 16)
(1 4)(2 11)(3 8)(5 9)(6 7)(10 15)(12 16)(13 14)
(1 5)(2 12)(3 6)(4 9)(7 8)(10 13)(11 16)(14 15)
(3 4 5 6 7 8 9)(10 11 12 13 14 15 16)
(3 4 6)(5 8 7)(10 11 13)(12 15 14)

G:=sub<Sym(16)| (1,2)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16), (1,3)(2,10)(4,8)(5,6)(7,9)(11,15)(12,13)(14,16), (1,4)(2,11)(3,8)(5,9)(6,7)(10,15)(12,16)(13,14), (1,5)(2,12)(3,6)(4,9)(7,8)(10,13)(11,16)(14,15), (3,4,5,6,7,8,9)(10,11,12,13,14,15,16), (3,4,6)(5,8,7)(10,11,13)(12,15,14)>;

G:=Group( (1,2)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16), (1,3)(2,10)(4,8)(5,6)(7,9)(11,15)(12,13)(14,16), (1,4)(2,11)(3,8)(5,9)(6,7)(10,15)(12,16)(13,14), (1,5)(2,12)(3,6)(4,9)(7,8)(10,13)(11,16)(14,15), (3,4,5,6,7,8,9)(10,11,12,13,14,15,16), (3,4,6)(5,8,7)(10,11,13)(12,15,14) );

G=PermutationGroup([[(1,2),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16)], [(1,3),(2,10),(4,8),(5,6),(7,9),(11,15),(12,13),(14,16)], [(1,4),(2,11),(3,8),(5,9),(6,7),(10,15),(12,16),(13,14)], [(1,5),(2,12),(3,6),(4,9),(7,8),(10,13),(11,16),(14,15)], [(3,4,5,6,7,8,9),(10,11,12,13,14,15,16)], [(3,4,6),(5,8,7),(10,11,13),(12,15,14)]])

G:=TransitiveGroup(16,712);

On 28 points - transitive group 28T44
Generators in S28
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 19)(9 20)(10 21)(11 15)(12 16)(13 17)(14 18)
(1 18)(2 19)(5 15)(7 17)(8 28)(11 24)(13 26)(14 27)
(1 18)(2 19)(3 20)(6 16)(8 28)(9 22)(12 25)(14 27)
(2 19)(3 20)(4 21)(7 17)(8 28)(9 22)(10 23)(13 26)
(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 9 11)(10 13 12)(15 19 20)(16 21 17)(22 24 28)(23 26 25)

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

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

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

G:=TransitiveGroup(28,44);

Polynomial with Galois group C2×AΓL1(𝔽8) over ℚ
actionf(x)Disc(f)
14T18x14+5x12-11x10-25x8+27x6+23x4-17x2+1-238·7312

Matrix representation of C2×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
,
 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 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
,
 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],[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],[0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0],[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] >;

C2×AΓL1(𝔽8) in GAP, Magma, Sage, TeX

C_2\times {\rm AGammaL}_1({\mathbb F}_8)
% in TeX

G:=Group("C2xAGammaL(1,8)");
// GroupNames label

G:=SmallGroup(336,210);
// by ID

G=gap.SmallGroup(336,210);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,2,116,2529,351,1900,856,767,1277]);
// Polycyclic

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

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