There are multiple artists with this name:
1) Alligator is an indie rock duo from Lyon, France, formed in 2011 by Elizabeth Hargrett (bass, vocals) and Lisa Duroux (drums, vocals). Facebook page: https://www.facebook.com/pages/ALLIGATOR/272848532737558
2) Alligator is a thrash metal band from Vigevano, Italy, founded in 1988. The band, heavily influenced by the Bay Area thrash, debuted in 1990 with the demo "Bog of Horrors". They have released 3 studio albums and disbanded after their last release in 1996.
3) Alligator is a nitzhonot DJ from Israel.