The coclass of a finite pgroup of order p^{n} and nilpotency class c is defined as n−c. This invariant of finite pgroups has been introduced by LeedhamGreen and Newman in LGN80 and it became of major importance in pgroup theory.
A first tool in the classification of all pgroups of coclass r is the coclass graph G(p,r). Its vertices are the isomorphism types of finite pgroups of coclass r. Two vertices G and H are joined by an edge if G is isomorphic to the quotient H/γ(H) where γ(H) is the last nontrivial term of the lower series of H.
Du Sautoy dS00 and Eick and LeedhamGreen ELG08 proved that G(p,r) contains certain periodic patterns. Eick and LeedhamGreen ELG08 define infinite coclass sequences of finite pgroups of coclass r which underpin this periodic pattern. In G(2,r) and G(3,1) almost all groups are contained in an infinite coclass sequence.
Eick and LeedhamGreen ELG08 also proved that the infinitely many pgroups in an infinite coclass sequence can be defined by a single parametrised presentation.
The first aim of this package is the definition of polycyclic parametrised presentations; these are parametrised presentations as defined by Eick and LeedhamGreen ELG08 and additionally they have various features of polycyclic presentations. Each such presentation defines all the infinitely many finite pgroups in an infinite coclass sequence.
We then provide some algorithms to compute with polycyclic parametrised presentations. In particular, we introduce a generalisation of the collection algorithm for polycyclic parametrised presentations. Based on this, we describe algorithms to compute polycyclic parametrised presentations for Schur extensions, for the Schur multiplicator and for some lowdimensional cohomology groups. We refer to EF11 for details on the underlying algorithms and further references.
Finally, we exhibit a database of polycyclic parametrised presentations for the infinite coclass families of the finite 2groups of coclass at most 2 and the finite 3groups of coclass 1.
In this section we describe the polycyclic parametrised presentations (pppresentations) for infinite coclass sequences.
Let (G_{x}  x ∈ N), where N denotes the natural numbers, be an
infinite coclass sequence; x is the parameter of this infinite coclass
sequence. Then every group G_{x} is an extension of a finite pgroup P
of order p^{n} by an abelian pgroup T_{x} of rank d. Furthermore, every
G_{x} has a polycyclic presentation (short pppresentation) on generators
g_{1}, …, g_{n}, t_{1}, …, t_{d} with relations of the form

We call such a pppresentation integral if all the padic numbers b_{k,l,m}, c_{k,l,m}, d_{k,l,m} are integers. Our algorithms introduced in this package compute with integral pppresentations only.
We call such an pppresentation consistent if for every x ∈ N the presentation is consistent as a polycyclic presentation; where we possibly reduce the exponents in the presentation modulo the relative orders of the generators.
In this section we recall briefly the method of EF11 to determine the Schur multiplicators of almost all groups G_{x} in an infinite coclass sequence.
Suppose we are given a consistent integral pppresentation F/R_{x} for the groups G_{x} in an infinite coclass sequence, where F is a free group and R_{x} is generated by parametrised relations as above. Note that the exponents in these relations depend on x, while the number of generators and the number of relations does not depend on the parameter.
Using this presentation we can define a parametrised presentation for the Schur extensions G_{x}^{*} = F/[F,R_{x}], corresponding to the parametrised presentation F/R_{x}. The next step is to find the isomorphism types of Y_{x} = R_{x}/[F,R_{x}] since M(G_{x}) ≅ (F′∩R_{x})/[F,R_{x}] are the torsion subgroups of Y_{x} as all G_{x} are finite pgroups.
Then Y_{x} = R_{x}/[F,R_{x}] are generated by certain socalled consistency relations. Using this we can compute the isomorphism types of Y_{x} and thus the isomorphism types of M(G_{x}) for almost all G_{x} in the chosen infinite coclass sequence.
From the parametrised presentation F/R_{x} we can see that the Abelian invariants are the same for all groups G_{x} in an infinite coclass sequence, and we can compute them. Using this and the computation of the Schur multiplicators one obtains H^{n}(G_{x},Z) and H^{n}(G_{x},GF(p)) for 0 ≤ n ≤ 2, where the G_{x} act trivially on Z and GF(p), respectively.
In this section we present the wellknown example of quaternion groups Q_{2x+3}. It is well known that they have a parametrised presentation of the following form:

Using this we can define the Schur extensions Q_{2x+3}^{*}

This yields M(Q_{2x+3}) = 1.
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