C₃⋊D₄₅ #63200

order	270
isoclinism	isoclinic class of C₃⋊D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₄₅
unambiguous semidirect product	C₅⋊(C₃⋊D₉)
unambiguous semidirect product	C₉⋊D₁₅

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₅×C₉	C₂
Frattini	C₃	C₃⋊D₁₅
socle	C₃²×C₅	D₃
LCS	G ▷ C₃×C₅×C₉
UCS	C₁

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 135

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

2,3^2,5,3|b^3=e^2,b^a=b^2e,c^a=c^2,d^a=d^4,e^a=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 18);