order	126
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×C₃×AGL(2,𝔽₃) (order 2592) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×C₃×S₄
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₃

Aliases

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

Splittings

Fitting, socle, Hall	C₃²×C₇ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₇

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

24 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 14);