C₂³×AGL(1,𝔽₇) #66747

order	336
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×AGL(1,𝔽₇)
unambiguous semidirect product	D₇⋊(C₂³×C₃)
unambiguous semidirect product	ΓL(1,𝔽₈)⋊C₂⁴
unambiguous semidirect product	(C₂×ΓL(1,𝔽₈))⋊C₂³

	subgroup	G/H
center, hypercenter	C₂³	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³

268 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 4 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 216);