Robin and I have been thinking a bit about the following problem. Consider an irrep of a group that is a unitary 1-design (and isn't also a unitary 2-design). For example, take a d-dimensional irrep of the Weyl-Heisenberg group mod d. We know that the action of this irrep on some vector will also produce a state 1-design. By analogy with SICPOVMs, we can imagine that the action could also produce a higher-order state design. We have taken to calling these vectors "voodoo states", since it seems to involve black magic to construct them. Robin and I are currently thinking about voodoo states for other groups, which might lead to some insight into SICs. More will be forthcoming.
Older stuff: Have a look at that:
Comment from David:
Steve refers to two papers:
 A. Nobs, Comment. Math. Helv. 51, (4) 46589 (German) (1976)
 A. Nobs and J. Wolfart, Comment. Math. Helv. 51, (4) 491526 (German) (1976)
which detail the structure of SL(2,Z_d). It's a very good and complete paper. I'm prepared to translate paragraphs should there be some interest. Look at the tables to get a rough idea of what you might want to know about. :-)