C2xAGammaL(1,8) - GroupNames

G = C₂×AΓL₁(𝔽₈) order 336 = 2⁴·3·7

Direct product of C₂ and AΓL₁(𝔽₈)

direct product, non-abelian, soluble, monomial, A-group

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 >

C₁

C₂

₇C₂

₂₈C₃

₈C₇

₇C₂²

₂₁C₂²

₂₈C₆

₈C₁₄

₈C₇⋊C₃

C₂³

₇C₂³

₇A₄

₂₈C₂×C₆

₈C₂×C₇⋊C₃

C₂⁴

₇C₂×A₄

F₈

₇C₂²×A₄

C₂×F₈

AΓL₁(𝔽₈)

C₂×AΓL₁(𝔽₈)

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

action	f(x)	Disc(f)
14T18	x¹⁴+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

Export

Subgroup lattice of C₂×AΓL₁(𝔽₈) in TeX
Character table of C₂×AΓL₁(𝔽₈) in TeX

G = C2×AΓL1(𝔽8) order 336 = 24·3·7

Direct product of C2 and AΓL1(𝔽8)

G = C₂×AΓL₁(𝔽₈) order 336 = 2⁴·3·7

Direct product of C₂ and AΓL₁(𝔽₈)