C₃⋊D₁₅ #782

order	90
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(2,𝔽₃) (order 8640) · Inn(G) = G · Out(G) = #985
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₁₅
unambiguous semidirect product	C₅⋊(C₃⋊D₃)

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

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 3 generators on 11 points:

(2,3)(4,5)(7,8)(10,11);(1,2,4,5,3)(9,10,11);(6,7,8)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(90, 9);