order	112
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = AΓL(1,𝔽₈) (order 168) · Inn(G) = AGL(1,𝔽₈) · Out(G) = C₃
holomorph	G ⋊ Aut(G) = C₂×#72214179
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×AGL(1,𝔽₈)
unambiguous semidirect product	C₂⁴⋊C₇

Splittings

Fitting, socle, Sylow	C₂⁴ ⋊ C₇
derived	C₂³ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₂	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₂⁴	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₂

17 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 14

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2)(4,5,6,8,10,7,9);(3,7)(4,5)(6,10)(8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 41);