order	144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×#120 (order 192) · Inn(G) = C₂² · Out(G) = C₂×C₃×D₄
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

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

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²
commutator	C₂	C₂×C₄×C₉
Frattini, socle, prime radical, cosocle	C₂²×C₃	C₂²×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

39 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 22);