#2069⋊C₃ #68037

order	384
isoclinism	isoclinic class of #2069⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Sylow	#2069 ⋊ C₃
derived	#338 ⋊ C₂×C₃

	subgroup	G/H
center, socle	C₂	#396⋊C₃
commutator, cosocle, hypocenter	#338	C₂×C₃
Frattini	C₂²×C₄	C₂×A₄
Fitting	#2069	C₃
hypercenter	C₄	#126⋊C₃
LCS	G ▷ #338
UCS	C₁ ◁ C₂ ◁ C₄

66 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:

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

Polycyclic presentation:

2,3,2^6|c^2=ef,d^2=efg,e^2=h,b^a=bfg,c^a=cf,d^a=dg,e^a=eh,c^b=cd,d^b=cfh,e^b=efg,f^b=fgh,g^b=f,d^c=deg,e^c=eh,f^c=fh,g^c=gh,e^d=eh,g^d=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 613);