order	252
isoclinism	isoclinic class of D₃×D₂₁
automorphism	Aut(G) = D₃²×AGL(1,𝔽₇) (order 1512) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₂₁
unambiguous semidirect product	(C₃×D₃)⋊D₇
unambiguous semidirect product	(D₃×C₇)⋊D₃

Splittings

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁

44 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)(5,6)(8,9)(10,11)(12,13);(2,3)(4,5,6)(7,8,10,12,13,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 36);