C₃⋊(#127⋊C₃) #63757

order	288
isoclinism	isoclinic class of C₃⋊(#127⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₄

Aliases

unambiguous semidirect product	C₃⋊(#127⋊C₃)
unambiguous semidirect product	#874⋊C₃

C₃×C₄² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄²	C₂×C₃
Frattini	C₂²	D₃×A₄
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ C₃×C₄²
UCS	C₁

59 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2,3,2^2,3,2^2|c^2=fg,d^2=g,c^a=cfg,d^a=dg,e^a=e^2,c^b=cd,d^b=cg,f^b=g,g^b=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 405);