C₃⋊(C₃⋊D₁₅) #63211

order	270
isoclinism	isoclinic class of C₃⋊(C₃⋊D₁₅)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃³

Aliases

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

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₅	C₂
LCS	G ▷ C₃³×C₅
UCS	C₁

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 135

A minimal faithful permutation representation — 4 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 29);