C₅⋊D₄₅ #90653

order	450
isoclinism	isoclinic class of C₅⋊D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product	C₅⋊D₄₅
unambiguous semidirect product	C₉⋊(C₅⋊D₅)

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅²×C₉	C₂
Frattini	C₃	C₃⋊(C₅⋊D₅)
socle	C₃×C₅²	D₃
LCS	G ▷ C₅²×C₉
UCS	C₁

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 225

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 18);