#392⋊C₃ #4635

order	192
isoclinism	isoclinic class of #392⋊C₃
automorphism	Aut(G) = C₂×#4626⋊C₂ (order 768) · Inn(G) = #127⋊C₃ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

#392 ⋊ C₃

	subgroup	G/H
center, hypercenter	C₂	#127⋊C₃
commutator, cosocle, hypocenter	#95	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#392	C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂

67 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

2,3,2^5|a^2=g,c^2=efg,d^2=eg,c^a=cef,d^a=de,c^b=d,d^b=cdf,e^b=f,f^b=ef,d^c=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 194);