D₃²×C₇ #6996

order	252
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₂×C₃×GO⁺(4,𝔽₂) (order 432) · Inn(G) = D₃² · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃²×C₇
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₇)
unambiguous semidirect product	(C₃⋊D₃)⋊(C₂×C₇)

Fitting, socle, Hall	C₃²×C₇ ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂²×C₇

	subgroup	G/H
center, hypercenter	C₇	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 35);