Q₈×C₉ #614

order	72
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×S₄ (order 144) · Inn(G) = C₂² · Out(G) = C₂×C₃×D₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×#5935
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₉

	subgroup	G/H
center	C₂×C₉	C₂²
commutator	C₂	C₂²×C₉
Frattini, socle, prime radical, cosocle	C₂×C₃	C₂²×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ G

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,4,6)(3,7,8,5)(9,10,12,11,13,15,14,16,17);(1,3,4,8)(2,5,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 11);