Note that PSL( 3, 4 ) has the same order as the alternating group of degree 8.
However, PSL( 3, 4 ) and Alt( 8 ) are not isomorphic. First, Alt( 8 ) has an element of order equal to 15.
Next, there is no element of order 15 in PSL( 3, 4 ).
This shows that there are two non-isomorphic simple groups of order 20160.
Several among the small projective special linear groups are isomorphic to alternating groups.