C₃⋊#1020 #64388

order	288
isoclinism	isoclinic class of C₃⋊#1020
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂³×C₃

Aliases

Fitting, socle	C₂⁴×C₃ ⋊ D₃
derived	C₃×C₂²⋊A₄ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₂²⋊A₄	C₂
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ C₃×C₂²⋊A₄
UCS	C₁

134 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

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

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

Polycyclic presentation:

2,3^2,2^4|b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,d^b=e,e^b=de,f^b=fg,g^b=f

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1036);