C₃×#10201 #10628808

order	768
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

528 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3189);