C₃×#24112 #10642719

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	C₃×#3490	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ G

1518 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)(6,11)(7,9)(8,10)(12,13)(14,16)(15,17)(18,19);(4,6,8,7)(5,9,10,11)(12,14)(13,16)(15,18)(17,19);(4,7,8,6)(5,9,10,11)(16,19);(5,10)(9,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17100);