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The Klein four-group is the direct product of two symmetric groups $S_2\times S_2$ with elements (here given in terms of cycles)

$$(1)(2)(3)(4),~(12)(34),~(13)(24),~(14)(23)$$

These cycles can be interpreted in terms of the individual permutation groups in $S_2\times S_2$ as an action of an operator $p$ that permutes pairs of elements:

$$(1)(2)(3)(4)=1\times 1~~~\text{ identity}\\(12)(34)=p\times 1~~~\text{exchange pairs }[1,3]\leftrightarrow[2,4]\\(13)(24)=1\times p~~~\text{exchange pairs }[1,2]\leftrightarrow[3,4]\\(14)(23)=p\times p~~~\text{exchange pairs }[1,3]\leftrightarrow[2,4]\text{ and then }[1,2]\leftrightarrow[3,4]$$

Now consider the generalization to $S_n\times S_m$ for some integer $n,m>1$ where the $S_n$ permutes collections of $m$ elements and the $S_m$ permutes collections of $n$ elements as demonstrated in the $n=m=2$ case above. Does this generalized group have a name? Has it been studied in the past? It might be a long shot, but is the cycle index polynomial for this group known?

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    @QiaochuYuan Unfortunately, this is not true. I asked a similar question in the following link and the difference is demonstrated there already in the case of $S_2\times S_2$. You will see that I have a construction algorithm for the cycle index polynomial implemented in an answer there, but I would like to verify it with some reference. http://mathematica.stackexchange.com/questions/137643/how-to-get-the-cycleindexpolynomial-of-direct-product-of-two-symmetric-groups – 2017-02-22
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    @QiaochuYuan The point is that a naive direct product $S_2\times S_2$ would have elements (1)(2)(3)(4), (12)(3)(4), (1)(2)(34), (12)(34), which is different from the ones we have in the Klein four-group for instance. – 2017-02-22
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    Sorry, I misunderstood your question. I don't understand your construction enough to see what the generalization you have in mind is. Can you clarify? (But even if I did, I don't expect there to be a name for the resulting permutation group.) – 2017-02-22
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    @QiaochuYuan I guess the easiest way to visualize the construction is to consider permutations of rows and columns of an $n\times m$ matrix. Group $S_n\times S_m$ acts on the $n m$ elements of the matrix by permuting rows and/or columns. This is different from an exterior direct product where one would consider $n+m$ elements instead and permute disjoint subsets of the $n$ and/or $m$ elements within themselves. – 2017-02-22
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    Okay, that's what I thought. Again, I don't think this permutation group has a name, but its cycle index polynomial shouldn't be too hard to figure out. – 2017-02-22

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