Consider the subgroup G of S_52 generated by the up and down riffle shuffle.

1. Prove that G acts imprimitively on the set of 52 cards, by considering the decomposition of {1,..,52} into the 26 sets {a,53-a}.

2. Deduce that the order of G is at most 26! * 2^26

3.(*) Determine whether G has order 26! * 2^26 or not.

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