#2091⋊C₃ #68039

order	384
isoclinism	isoclinic class of #2017⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Sylow	#2091 ⋊ C₃
derived	#338 ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₄	#127⋊C₃
commutator, cosocle, hypocenter	#338	C₂×C₃
Frattini	C₂²×C₄	C₂×A₄
Fitting	#2091	C₃
socle	C₂	#399⋊C₃
LCS	G ▷ #338
UCS	C₁ ◁ C₄

67 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:

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

Polycyclic presentation:

2,3,2^6|c^2=e,d^2=efg,e^2=h,b^a=bg,c^a=cf,d^a=dg,c^b=cdefh,d^b=cgh,e^b=eg,f^b=fg,g^b=fh,d^c=def,g^c=gh,e^d=eh,f^d=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 615);