#2401⋊C₃ #73189

order	384
isoclinism	isoclinic class of #396⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, hypercenter	C₄	#126⋊C₃
commutator, cosocle, hypocenter	#95	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2401	C₃
prime radical	#396⋊C₃	C₂
LCS	G ▷ #95
UCS	C₁ ◁ C₄

107 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:

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

Polycyclic presentation:

2^2,3,2^5|b^2=f,d^2=fg,e^2=fgh,c^a=cg,d^a=dgh,e^a=eh,c^b=ch,d^c=defh,e^c=d,g^c=h,h^c=gh,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5765);