  We introduce twisted unitary $t$-groups, a generalization of unitary
$t$-groups under a twisting by an irreducible representation. We then apply
representation theoretic methods to the Knill-Laflamme error correction
conditions to show that twisted unitary $t$-groups automatically correspond to
quantum codes with distance $d=t+1$. By construction these codes have many
transversal gates, which are naturally fault tolerant.
