#396⋊C₃ #4638

order	192
isoclinism	isoclinic class of #396⋊C₃
automorphism	Aut(G) = C₂×#560⋊C₃ (order 384) · Inn(G) = #126⋊C₃ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting, Sylow	#396 ⋊ C₃
derived	#95 ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₂	#126⋊C₃
commutator, cosocle, hypocenter	#95	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#396	C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂

51 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:

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

Polycyclic presentation:

2,3,2^5|c^2=ef,d^2=eg,c^a=ce,d^a=dfg,c^b=d,d^b=cdf,e^b=fg,f^b=ef,d^c=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 197);