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I currently have a set of reflections/rotations that I believe is a generating set of the symmetry group of a regular octahedron. However, I do no know how to show this set will generate the entire group without iterating through the entire group. The 2nd part of my question is to find a set for an n-dimensional cube which seems to have some dual relationships with the octahedron so I feel like iterating through all the 48 elements is not the desired answer.

Thanks for your help!

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 Can you get ever signed permutation matrix this way? The question is more appropriate for math.stackexchange. – Scott Carter Feb 10 at 4:47
The "algebraic geometry" tag should be removed or explained. – Vladimir Dotsenko Feb 10 at 5:35
When/if you ask on math.SE: give details about what your generating set is. In general, there is not much more to be done that iterate through the generated subgroup, but in a concrete situation such as yours he geometry can give information which allows you to avoid that. Otherwise, your question is «how can I check that a subset generates my group?» which doesn't have an answer you like. – Mariano Suárez-Alvarez Feb 10 at 5:59
You're right Vlad, I removed it. The generating set is actually one we have to come up with ourselves so I was hoping there are some properties on the octahedral that will make iterating group elements easier. Thanks Scott, I probably will post on stackexchange, it does seem more suitable. – Azhuang Feb 10 at 6:10
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 If you foresee yourself doing a lot of these computations, you may want to use a program like GAP. – S. Carnahan♦ Feb 10 at 9:08

closed as too localized by Misha, Scott Carter, Vladimir Dotsenko, Mariano Suárez-Alvarez, S. Carnahan♦ Feb 10 at 9:07

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