order	144
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 432) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×(C₂²⋊C₉)⋊C₂
unambiguous semidirect product	(C₂²⋊C₉)⋊C₂²
unambiguous semidirect product	(C₂×C₂²⋊C₉)⋊C₂

Splittings

C₂²⋊C₉ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂²
Frattini	C₃	C₂×S₄
Fitting, socle	C₂³×C₃	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 18

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 109);