C₃×#1020 #64387

order	288
isoclinism	isoclinic class of #1020
automorphism	Aut(G) = C₂×(#1020⋊C₃)⋊C₂ (order 1152) · Inn(G) = #1020 · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³

Aliases

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

	subgroup	G/H
center, hypercenter	C₃	#1020
commutator, cosocle, hypocenter	C₂²⋊A₄	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ C₂²⋊A₄
UCS	C₁ ◁ C₃

119 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1035);