#399⋊C₃ #4639

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

Fitting, Sylow	#399 ⋊ C₃
derived	#95 ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₂	#127⋊C₃
commutator, cosocle, hypocenter	#95	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#399	C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂

51 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 20 points:

(2,3,5)(6,8,7)(9,10,13)(11,19,18,16,14,20)(12,15,17);(1,5,4,7)(2,8,6,3)(9,16,12,11)(13,18,17,20)

Polycyclic presentation:

2,3,2^5|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, 198);