I'm a fan of the isomorphism between $PSL_2 (F_7)$ and $GL_3 (F_2)$ (two nice descriptions of the simple group of order 168). Years ago, Richard Guy asked me if I knew an explicit map, and I didn't. But recently one was given in the Math Monthly:
MR2572107 Brown, Ezra; Loehr, Nicholas Why is ${\rm PSL}(2,7)\cong{\rm GL}(3,2)$? Amer. Math. Monthly 116 (2009), no. 8, 727--732.