between the sequential fourth generation and the extra vector-like leptons are allowed. The lepton flavor violating phenomenology in this model is very rich and needed to be studied further. Before we close, we note that $A_{4}$ extended in the binary tetrahedral group $T'$ can provide a model of both the quark and lepton sectors [@Tprime]. The three generation $T'$ model has calculable Cabibbo angle as well as other attractive features [@Tprime]. There is an analogous extension of the $A_{5}$ to the binary icosahedral group $I'$, where the attractive features of the $T'$ model can be utilized and four families of quarks and leptons can be accommodated simultaneously [@CKY2].

This work was supported in part by the US DOE grant DE-FG05-85ER40226, National Science Council of Taiwan under Grant No. 98-2112-M-001-014-MY3 and the National Center for Theoretical Sciences of Taiwan (NCTS). CSC would like to thank L. F. Li for useful discussion. TWK is grateful for the hospitality of the Physics Division of NCTS where this work was initiated.

Discrete Symmetry Group $A_{5}$ and the Icosahedron
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![The regular icosahedron.[]{data-label="fig:icosahedron"}](icosahedral.eps){width="30.00000%"}

$A_{5}$ is a discrete symmetry group of even permutation of five objects. Its order, the number of elements, is equal to $(5!)/2=60$, 