C₃×D₉ #267

order	54
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂×AGL(1,ℤ/9) (order 108) · Inn(G) = D₉ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = D₃×#92265⋊C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

C₃×D₉

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

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

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 3);