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AΓL(1,𝔽₈) #4273

order	168
isoclinism	isoclinic class of AΓL(1,𝔽₈)
automorphism	Aut(G) = AΓL(1,𝔽₈) (order 168) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = AΓL(1,𝔽₈)²
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

affine general semilinear	AΓL(1,𝔽₈)
unambiguous semidirect product	AGL(1,𝔽₈)⋊C₃

Splittings

Fitting, socle, Sylow	C₂³ ⋊ ΓL(1,𝔽₈)
derived, Hall	AGL(1,𝔽₈) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	AGL(1,𝔽₈)	C₃
Fitting, socle	C₂³	ΓL(1,𝔽₈)
LCS	G ▷ AGL(1,𝔽₈)
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

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

Polycyclic presentation:

3,7,2^3|b^a=b^2,c^a=ce,d^a=cd,e^a=c,c^b=d,d^b=e,e^b=ce

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	7	24	24	28	28	28	28
order	1	2	7	7	3	3	6	6
χ₁	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	3	$\frac{- 1 - i \sqrt{7}}{2}$	$\frac{- 1 + i \sqrt{7}}{2}$	0	0	0	0
χ₅	3	3	$\frac{- 1 + i \sqrt{7}}{2}$	$\frac{- 1 - i \sqrt{7}}{2}$	0	0	0	0
χ₆	7	-1	0	0	1	1	-1	-1
χ₇	7	-1	0	0	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₈	7	-1	0	0	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	C₂×AΓL(1,𝔽₈), C₃×AΓL(1,𝔽₈), C₂²⋊AΓL(1,𝔽₈), …
maximal normal subgroup of	C₂×AΓL(1,𝔽₈), C₃×AΓL(1,𝔽₈), C₅×AΓL(1,𝔽₈), …
maximal quotient of	C₂×AΓL(1,𝔽₈), AGL(1,𝔽₈)⋊C₉, C₃×AΓL(1,𝔽₈), …
automorphism group of	AGL(1,𝔽₈), C₂×AGL(1,𝔽₈), AΓL(1,𝔽₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 43);