info.7

5. Format of PCP files.

Format of files containing PCPs of p-groups.

Abbreviations:  dM= dual of module M, sd = semidirect product,
                  tp-P= tensor product over P.

Types of Files:

  (i) Files gpname.pcp (output of pcrun).
    Gives PCP of original p-group P.

  (ii) Files gpname.sc (output of scrun).
    Gives list of generators and PCP of Sylow intersections Q = P ^ gPg-.

  (iii)  Files gpname.cov (output of nqmrun).
    Gives PCP of a central extension E of an abelian group  M  by P.

  (iv) Files gpname.ch1  (output of nqrun with -1 set).
    Gives PCP of a central extension E of an elementary abelian p-group M1 by
    P sd dM, with M1 <= [P,dM]. M1 is isomorphic to the dual of a subgroup of
    H^1(P,M).

  (v) Files gpname.ch2  (output of nqrun with -1  not set).
    Gives PCP of a central extension E of an elementary abelian p-group M2 by
    FE sd dM.  FE is itself an extension of an elementary abelian p-group FM
    by P, with FM <= Frattini(FE), and M2 = FM tp-P dM. H^2(P,M) is the dual
    of a subgroup M2a of M2, whose generators are given, and finally H^2(G,M)
    is the dual of M2a/M2b, where generators of M2b are also listed.

Some generalities:
  Except in files gpname.sc, Line 1 contains 6 numbers:
  prime,exp,facexp,no,class,mult;
  exp=exponent of P (cases (i),(iii),(iv)) of Q (case (ii)) or of FE (case (v)).
  facexp=exponent of P (case (i),(iv),(v)) or no. of gens of M (case (ii)).
  mult=1 for computation of Schur multipliers, otherwise 0.
  no is explained below.
  Line 2 contains weights of generators 1,2,3,...
  Lines 3 and 4 contain d1 and d2 for these generators, where the definition
  of generator i is given by the commutator [d1(i),d2(i)] if d1(i) != d2(i),
  and by the p-th power of d1(i) otherwise;
  The following lines contain the values of the commutators and powers in the
  order [2,1],[3,1],[3,2],[4,1],...,1^p,2^p,...
  In general, generators with values higher than no (in line 1) are central of
  order p, and so commutators [j,i] and powers j^p are given only for j<=no.
  In case (v), generators of FE with values higher than facexp have order p,
  and commute with each other, so the corresponding trivial powers and
  commutators are not listed.
  There is one line for each such commutator or power. The first number is zero
  if this is not a definition, and k if it is the definition of generator k.
  The second number is the length of the expression, and the expression itself
  follows, as a generator-power string.
  In future, we shall call such a sequence a dgps (definition-generator-power
  string).
  e.g.  5  4 3 2 5 1  means that the expression equals 3^2.5^1, and is the
  definition of generator 5.

Exceptions and specific cases:

  Case (i).  This is simply the PCP for P. no=exp-1.

  Case (ii).    This file is in blocks, each one corresponding to a Sylow
    intersection Q = P ^ gPg-, for some  g. The PCP is for Q in each case.
    The case g in N(P) (Q=P) is treated as a special case.
    when a variable norm is set true.
    The first line in each block contains the exponent of Q, and,if mult=0,
    the class of Q. (PCP with full information for Q, with wt,d1,d2 is only
    required if mult=0.) The file ends with a line containing -1 only.
    If mult=0, line 2 contains values of wt, for generators of Q.
    If not norm, then there follows a list of the generators of Q, given as
    generator-power strings (preceded by length of string) in the generators
    of P, each on a new line. (If norm, then the generators of Q are the
    same as those of  P.)
    In any case, this is followed by a list of the conjugates of these
    generators under g, in the same format.
    If norm is true, then there is no further output in this block.
    Otherwise, if mult=0, d1 and d2 follow, and then, in any case, comes the
    PCP for Q, with one dgps for each expression [2,1],...
    Powers are only given when mult=0 (they are not needed otherwise).

  Case (iii).  Let us call the generators of P  1,2,3,..., and those of  M
    ng1,ng2,...
    wt is given only for generators of P, but d1 and d2 are given for P and M,
    in the order 1,2,3,...,ng1,ng2,... Of course, d1[ngi]=x and d2[ng2]=y
    means that ngi has definition [x,y], where x and y are gens of P.
    d2 is follwed by an extra line, giving the orders of the generators of
    M. These will all be powers of prime.
    The commutators and powers [2,1],..., 1^p,... follow as usual. Each such
    line contains two dgps. The first gives the value in P, and the second
    gives the element of M with which this must be multiplied to give the
    value in E. The definition no in the second dgps is given as exp+i for ngi.

    Cases (iv),(v). An important number dim (the dimension of the module  M)
    is not given in the file (it comes from the matrix files), but its value
    is implicitly required.
    In both cases, the elements of the base of the underlying vector space
    of dM are numbered exp+1,exp+2,... , and the PCP of the action of P on dM
    is included.
    wt,d1 and d2 are given for generators 1,2,..,exp,exp+1,...,exp+dim.
    For ch1, the PCP of P follows, and, for ch2, the PCP of FE.
    There follows a number nng, the number of generators of M1 or M2
    in cases ch1 and ch2 respectively.
    The definitions nd1 and nd2 of these new generators follow on the next two
    lines. nd1[i]=x and nd2[i]=y means that generator i of M1 or M2 has
    definition [x,y], where x is a gen in dM, and y is a gen in P (ch1)
    or FM (ch2).
    This is followed by the PCP of the action of P (ch1) or FE (ch2) on dM.
    The order is [exp+1,1],[exp+1,2],...,[exp+2,1],[exp+2,2],...,[exp+dim,exp],
    each on a new line.
    In case ch1, each such line contains two dgps components, one in dM itself,
    and one in the gens of M1. In case ch2, there is only one
    component in each case, which is in dM for [exp+i,j] with j<=facexp, and
    in the gens of M2 for j>facexp.
    In case (v), the lists of generators of the subgroups M2a and M2b of M2
    follow, preceded by their numbers chpdim and chsdim. Each such
    generator occupies one line. The first number is a characteristic number
    with which it can be recognized from the generator of M2 with this number.
    This is followed by the element itself, as a generator-power string in the
    generators of M2, preceded by its length.