C₃×#23595 #10642202

order	768
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center	C₂×C₃×C₄	#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
socle	C₂²×C₃	C₂×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

1616 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16583);