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Is there a way to fully specify the ordering of elements in an algebraic group? (This ordering affects the output of several other group-theoretical operations.)

For example, how can I set the ordering of elements in SymmetricGroup[3] to be precisely this

Cycles[{{}}],
Cycles[{{1, 2, 3}}],
Cycles[{{1, 3, 2}}],
Cycles[{{1, 2}}],
Cycles[{{1, 3}}],
Cycles[{{2, 3}}]

The docs say that one can control the ordering of group elements by specifying a suitable value for the GroupActionBase option, but I have not been able to come up with a value for this option that will result in the ordering shown above.

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I think you're seeing the result of MM's sort heuristics: "shorter" things get sorted to the front, caeteris paribus. If what you want is just the list from GroupElements in that kind of order, perhaps SymmetricGroup[3] // GroupElements // Join[{First[#]}, SortBy[Rest[#], {-Length[#[[1, 1]]]} &]] & might serve. –  rasher Jan 28 at 12:43
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