C₃×#24350 #10642957

order	768
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

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

1462 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)(4,5,8,11)(6,10,7,9)(12,13)(14,16)(15,17)(18,19);(4,6,8,7)(5,9,11,10)(12,14)(13,16)(15,18)(17,19);(16,19);(4,7,8,6)(5,10,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17338);