#2760⋊C₃ #73200

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, hypercenter	C₂	#560⋊C₃
commutator, cosocle, hypocenter	#95	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2760	C₃
prime radical	#392⋊C₃	C₂
LCS	G ▷ #95
UCS	C₁ ◁ C₂

115 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:

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

Polycyclic presentation:

2^2,3,2^5|a^2=fg,b^2=fg,d^2=h,e^2=f,b^a=bfg,d^a=dfh,e^a=efgh,c^b=cg,d^b=dfgh,e^b=eg,d^c=efgh,e^c=de,f^c=gh,g^c=fh,h^c=f,e^d=efg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5776);