C₂×C₃×AGL(1,𝔽₈) #66744

order	336
isoclinism	isoclinic class of AGL(1,𝔽₈)
automorphism	Aut(G) = C₂×AΓL(1,𝔽₈) (order 336) · Inn(G) = AGL(1,𝔽₈) · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₃×AGL(1,𝔽₈)
unambiguous semidirect product	(C₂⁴×C₃)⋊C₇

	subgroup	G/H
center, hypercenter	C₂×C₃	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₂×C₃

34 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 213);