A multimagic is a square, where in addition, the rows, columns and diagonals have the same sum, when all the numbers are taken to the second, third, fourth or even higher power. The first multimagic square is bimagic and was published 1890 by G. Pfeffermann.
A trimagic square is a bimagic square, where in addition, the rows, columns and diagonals have the same sum, when all the numbers are taken to the third power. The first trimagic square was given by Gaston Tarry in 1905 vor. This square has order 128.
You will find more information about tetra-, penta- and hexamagic squares on the multimagic pages of Christian Boyer.