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For distinct primes $p$ and $q$, how many nonabelian groups up to isomorphism are of the order $p^4*q^4$?

We can say that there are nontrivial subgroups with cardinality $p,p^2,p^3,p^4,p*q,..,q^4$. Since we are dealing with nonabelian groups, we can't say a group with cardinality $p^4*q^4$ is isomorpic to $$Z_(p^4*q^4)$$. Should I consider $$U_(p^4*q^4)$$ ?

anomaly
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Ninja
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  • Are you sure you really want to find all non-abelian groups of order $p^4q^4$, not just for the abelian ones? If yes, you could get acquainted with their $p$- rsp. $q$-Sylow subgroups by looking at the links given in the comments to these questions http://math.stackexchange.com/questions/1095008/reference-request-groups-of-order-p4 and http://math.stackexchange.com/questions/1162977/groups-of-order-p4 . – j.p. Mar 17 '15 at 17:35
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    I'd be surprised if anyone knows the answer to this for general primes $p$ and $q$. Is the answer even known for easier cases, such as $pq^3$ or $p^2q^2$? – James Mar 17 '15 at 17:46
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    Groups of orders like $pq^3$ and $p^2q^2$ were done at the beginning of the 20th century, see for example the introduction of http://www.ams.org/journals/tran/1906-007-01/S0002-9947-1906-1500737-3/S0002-9947-1906-1500737-3.pdf – verret Mar 18 '15 at 03:08

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