C₃⋊(C₅⋊D₅) #3881

order	150
isoclinism	isoclinic class of C₃⋊(C₅⋊D₅)
automorphism	Aut(G) order 72000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product	C₃⋊(C₅⋊D₅)
unambiguous semidirect product	C₅⋊D₁₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²	C₂
LCS	G ▷ C₃×C₅²
UCS	C₁

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 75

A minimal faithful permutation representation — 3 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 12);