#2030⋊C₃ #68034

order	384
isoclinism	isoclinic class of #2029⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, socle, hypercenter	C₂²	#120⋊C₃
commutator, cosocle, hypocenter	#543	C₂×C₃
Frattini	C₂⁴	C₂×A₄
Fitting	#2030	C₃
prime radical	#543⋊C₃	C₂
LCS	G ▷ #543
UCS	C₁ ◁ C₂²

85 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2,3,2^6|a^2=h,c^2=g,d^2=g,c^a=ce,d^a=dfh,c^b=cd,d^b=cgh,e^b=ef,f^b=eh,d^c=dg,f^c=fh,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 610);