Danny, thank you for the quick and very effective response.

This got me thinking about the generic way.

Clearly, we need some definition of the group, as applied to the representation, in my case:

d4Group = PermutationGroup[{Cycles[{{1, 3, 9, 7}, {2, 6, 8, 4}}], Cycles[{{2, 6}, {4, 8}, {1, 9}}] }]

Then we can generate the orbits for each permutation and just keep the distinct ones:

allDistinctPermutations = First /@ Union[ GroupOrbits[ d4Group, Permutations[Range[9]]]]

the whole thing only takes 40% longer than the specific Select

thank you Danny,

Luc