C₃²×AGL(1,𝔽₇) #67394

order	378
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃²×AGL(1,𝔽₇)
unambiguous semidirect product	D₇⋊C₃³
unambiguous semidirect product	ΓL(1,𝔽₈)⋊(C₂×C₃²)
unambiguous semidirect product	(C₃×D₇)⋊C₃²

	subgroup	G/H
center, hypercenter	C₃²	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂×C₃³
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 47);