C₃×#23430 #10642037

order	768
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

2706 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16418);