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The problem of existence of symmetric informationally-complete positiveoperator-valued measures (SICs for short) in every dimension is known asZauner's conjecture and remains open to this day. Most of the known SICexamples are constructed as an orbit of the Weyl-Heisenberg group action. Itappears that in these cases SICs are invariant under so-called canonical order3 unitaries, which define automorphisms of the Weyl-Heisenberg group. In thisnote we show that those order 3 unitaries appear in projective unitaryrepresentations of the triangle group $(3,3,3)$. We give a full description ofsuch representations and show how it can be used to obtain results about thestructure of canonical order 3 unitaries. In particular, we present analternative way of proving the fact that any canonical order 3 unitary isconjugate to Zauner's unitary if dimension $d>3$ is prime.

SICs and the triangle group (3,3,3)