order	144
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂×C₃×S₄ (order 144) · Inn(G) = A₄ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72268997
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×Q₈⋊C₉
unambiguous semidirect product	(C₂×Q₈)⋊C₉

Splittings

derived	Q₈ ⋊ C₂×C₉
Sylow	C₂×Q₈ ⋊ C₉

	subgroup	G/H
center, socle, prime radical, hypercenter	C₂²×C₃	A₄
commutator, hypocenter	Q₈	C₂×C₉
Frattini	C₂×C₃	C₂×A₄
Fitting	C₂×C₃×Q₈	C₃
cosocle	C₃×Q₈	C₂×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₃

29 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2)(4,5,7)(8,10,9)(11,15,19,13,18,14,16,12,17);(3,7,6,9)(4,10,8,5)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 35);