ΓL(1,𝔽₈)⋊C₈ #4231

order	168
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = C₂²×AGL(1,𝔽₇) (order 168) · Inn(G) = AGL(1,𝔽₇) · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	ΓL(1,𝔽₈)⋊C₈
unambiguous semidirect product	(C₇⋊C₈)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₄	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₈
Fitting, cosocle	C₄×C₇	C₂×C₃
socle	C₂×C₇	C₃×C₄
prime radical	C₂×ΓL(1,𝔽₈)	C₄
LCS	G ▷ C₇
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

(2,6,5,3,4,7)(8,9,10,12,11,13,14,15);(1,2,5,6,4,7,3)

Polycyclic presentation:

2,3,2^2,7|a^2=c,c^2=d,e^a=e^6,e^b=e^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 1);