C₃×#52103 #10670710

order	768
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
socle	C₂³×C₃	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

3356 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 45091);