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1 Enumerating groups of order p^nq for n ≤ 5
 1.1 Calculating the generalized PORC polynomials
 1.2 Evaluating the generalized PORC polynomials

1 Enumerating groups of order p^nq for n ≤ 5

The package pnqporc provides the methods to calculate generalized PORC polynomials in the indeterminates p and q enumerating the number of isomorphism types of groups of order p^nq for n ≤ 5 and different primes p and q.

These generalized PORC polynomials can contain expressions of the form d(s,t). When evaluating, the value of d(s,t) is 1 if s ∣ t and 0 otherwise.

Furthermore, the generalized PORC polynomials can contain expressions of the form g(p^m-1,q). When evaluating, the value of g(p^m-1,q) is 1 if m is minimal such that p^m-1 ∣ q and 0 otherwise.

For more details, we refer to the paper [EM17].

1.1 Calculating the generalized PORC polynomials

The following functions compute generalized PORC polynomials enumerating the number of isomorphism types of groups of order p^nq for n ≤ 5 (or certain subsets).

1.1-1 NumberPNQ
‣ NumberPNQ( n )( operation )

Given an integer n ≤ 5, this returns a generalized PORC polynomial in p and q enumerating the number of isomorphism types of groups of order p^nq.

1.1-2 NumberPNQNormalPSylow
‣ NumberPNQNormalPSylow( n )( operation )

Given an integer n ≤ 5, this returns a generalized PORC polynomial in p and q enumerating the number of isomorphism types of non-nilpotent groups of order p^nq with normal Sylow p-subgroup.

1.1-3 NumberPNQNormalQSylow
‣ NumberPNQNormalQSylow( n )( operation )

Given an integer n ≤ 5, this returns a generalized PORC polynomial in p and q enumerating the number of isomorphism types of non-nilpotent groups of order p^nq with normal Sylow q-subgroup.

1.1-4 NumberPNQExceptional
‣ NumberPNQExceptional( n )( operation )

Given an integer n ≤ 5, this returns a generalized PORC polynomial in p and q enumerating the number of isomorphism types of non-nilpotent groups of order p^nq, which do not contain a normal Sylow subgroup.

1.1-5 NumberPNQNilpotent
‣ NumberPNQNilpotent( n )( operation )

Given an integer n ≤ 5, this returns a generalized PORC polynomial in p and q enumerating the number of isomorphism types of nilpotent groups of order p^nq. This coincides with the number of isomorphism types of groups of order p^n.

1.2 Evaluating the generalized PORC polynomials

1.2-1 EvalPNQPorcPoly
‣ EvalPNQPorcPoly( poly, p, q )( operation )

This evaluates a generalized PORC polynomial returned by one of the functions in the previous section at the given primes p and q.

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