C₃×#25563 #10644170

order	768
nilpotency class	3
isoclinism	isoclinic class of #24404
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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁵×C₃ ◁ G

1608 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)(6,10)(7,9)(8,11)(12,13)(14,16)(15,17)(18,19);(4,6,7,11)(5,8,9,10)(12,14)(13,16)(15,19)(17,18);(12,13)(14,18,19,16)(15,17);(5,9)(8,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 18551);