C₂×D₃×D₉ #6264

order	216
isoclinism	isoclinic class of D₃×D₉
automorphism	Aut(G) = C₂²×D₃×AGL(1,ℤ/9) (order 1296) · Inn(G) = D₃×D₉ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₃×D₉
unambiguous semidirect product	(C₃×D₉)⋊C₂²
unambiguous semidirect product	(D₃×C₉)⋊C₂²
unambiguous semidirect product	(C₃⋊D₉)⋊C₂²

Fitting	C₂×C₃×C₉ ⋊ C₂²
derived, Sylow	C₃×C₉ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂	D₃×D₉
commutator, cosocle, hypocenter	C₃×C₉	C₂³
Frattini	C₃	C₂×D₃²
Fitting	C₂×C₃×C₉	C₂²
socle	C₂×C₃²	C₂×D₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂

106 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 101);