#393⋊C₃ #4636

order	192
automorphism	Aut(G) = C₂×#4625⋊C₂ (order 768) · Inn(G) = #120⋊C₃ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

Fitting, Sylow	#393 ⋊ C₃
derived	C₂²×Q₈ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₂	#120⋊C₃
commutator, cosocle, hypocenter	C₂²×Q₈	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#393	C₃
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

57 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 24

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

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

Polycyclic presentation:

2,3,2^5|c^2=g,d^2=g,c^a=ce,d^a=df,c^b=d,d^b=cd,e^b=f,f^b=ef,d^c=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 195);