#2062⋊C₃ #68036

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	#2062	C₃
prime radical	#543⋊C₃	C₂
LCS	G ▷ #543
UCS	C₁ ◁ C₂²

89 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:

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

Polycyclic presentation:

2,3,2^6|a^2=g,c^2=g,d^2=g,c^a=ce,d^a=df,c^b=cdg,d^b=cg,e^b=efh,f^b=e,d^c=dg,f^c=fh,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 612);