#401⋊C₃ #4640

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) = C₁

Aliases

#401 ⋊ C₃

	subgroup	G/H
center, hypercenter	C₂	#126⋊C₃
commutator, cosocle, hypocenter	#95	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#401	C₃
prime radical	#95⋊C₃	C₂
LCS	G ▷ #95
UCS	C₁ ◁ C₂

33 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 24 points:

(1,6,19,8,3,15,5,17,9,2,12,22)(4,20,18,7,11,24,14,16,21,13,10,23);(1,4)(2,13)(5,14)(6,17)(7,8)(9,21,19,18)(10,11)(15,23,22,24)

Polycyclic presentation:

2,3,2^5|a^2=g,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, 199);