C₃×#24170 #10642777

order	768
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

1162 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)(6,10)(7,8)(9,11)(12,13)(14,17)(15,18)(16,19);(5,8)(9,10)(12,13)(14,18)(15,17)(16,19);(13,17)(14,15)(18,19);(4,6,7,11)(5,9,8,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17158);