C₃²×#125 #64174

order	288
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

168 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 822);