#1020⋊C₃ #64377

order	288
isoclinism	isoclinic class of #1020⋊C₃
automorphism	Aut(G) = (#1020⋊C₃)⋊C₂ (order 576) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

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

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

46 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1025);