D₄×C₉ #613

order	72
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×D₄ (order 48) · Inn(G) = C₂² · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×#453
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₉
unambiguous semidirect product	C₄⋊(C₂×C₉)

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 10);