order	378
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₂×C₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 648) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×(C₃⋊D₃)⋊C₃
unambiguous semidirect product	(C₃⋊D₃)⋊(C₃×C₇)

Splittings

	subgroup	G/H
center, hypercenter	C₇	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂×C₃×C₇
Frattini	C₃	C₃×D₃×C₇
Fitting	C₇×H(3,𝔽₃)	C₂
socle	C₃×C₇	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,4,5,6,7)(9,14,16,11,12,15)(10,13);(8,9,11)(10,12,15)(13,16,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 34);