#2032⋊C₃ #68035

order	384
automorphism	Aut(G) = #73210 (order 384) · Inn(G) = #393⋊C₃ · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting, Sylow	#2032 ⋊ C₃
derived	#543 ⋊ C₂×C₃

	subgroup	G/H
center, socle	C₂	#393⋊C₃
commutator, cosocle, hypocenter	#543	C₂×C₃
Frattini	C₂⁴	C₂×A₄
Fitting	#2032	C₃
hypercenter	C₂²	#120⋊C₃
LCS	G ▷ #543
UCS	C₁ ◁ C₂ ◁ C₂²

83 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2)(3,11,12,6,4,16)(5,8)(7,9,15,13,14,10);(1,9)(2,16,8,14)(3,4)(5,12)(6,7,15,11)(10,13)

Polycyclic presentation:

2,3,2^6|c^2=g,d^2=g,c^a=ceh,d^a=df,g^a=gh,c^b=dg,d^b=cdgh,e^b=f,f^b=ef,d^c=dg,e^c=eh,f^c=fh,f^d=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 611);