C₂×C₄×C₉ #612

order	72
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×D₄ (order 48) · Inn(G) = C₁ · Out(G) = C₂×C₃×D₄
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×B(4,𝔽₂)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

abelian	C₂×C₄×C₉
multiplicative	(ℤ/95)ˣ, (ℤ/111)ˣ, (ℤ/135)ˣ, (ℤ/148)ˣ, (ℤ/190)ˣ, (ℤ/222)ˣ, (ℤ/270)ˣ

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 9);