order	216
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 432) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

	subgroup	G/H
center, hypercenter	C₃	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂×C₃
Frattini	C₃	C₃×S₄
Fitting, socle	C₂²×C₃²	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₃

41 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,4,3)(5,6,7)(9,11)(10,14)(12,13)(15,16);(2,3,4)(8,16,11,14,13,12,10,9,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 91);