order	216
isoclinism	isoclinic class of D₃×C₃⋊D₃
automorphism	Aut(G) = C₂²×D₃×AGL(2,𝔽₃) (order 10368) · Inn(G) = D₃×C₃⋊D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

Splittings

C₃³ ⋊ C₂×C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₂×C₄
Fitting, socle, cosocle	C₂×C₃³	C₂²
prime radical	C₂×C₃×C₃⋊D₃	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

120 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 125);