#2644⋊C₃ #73194

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center	C₂	C₂×#126⋊C₃
commutator, cosocle, hypocenter	#95	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2644	C₃
hypercenter	C₄	#126⋊C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂ ◁ C₄

105 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:

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

Polycyclic presentation:

2^2,3,2^5|b^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cgh,d^a=dgh,e^a=efh,d^c=degh,e^c=dh,g^c=h,h^c=fgh,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5770);