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This note contains some examples of hyperk\"ahler varieties $X$ having agroup $G$ of non-symplectic automorphisms, and such that the action of $G$ oncertain Chow groups of $X$ is as predicted by Bloch's conjecture. The examplesrange in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ isa Calabi-Yau variety which has interesting Chow-theoretic properties; inparticular, the variety $Y$ satisfies (part of) a strong version of theBeauville-Voisin conjecture.

Algebraic cycles on some special hyperk\"ahler varieties