In this chapter we describe how the consistent pppresentations of infinite coclass sequences can be used to compute a pppresentation for the corresponding Schur extensions (see EF11).
For a group G = F/R the Schur extension H is defined as H = F/[F,R] (see EN08).
So for a parameter x that can take values in the positive integers, let (G_{x} = F/R_{x}  x ∈ N), for N the positive integers, describe an infinite coclass sequence of finite pgroups G_{X} of coclass r. Then for each value for the parameter x, the group G_{x} has a consistent polycyclic presentation with generators g_{1}, ·.·, g_{n}, t_{1}, ·.·, t_{d} and relations

Then we compute a consistent pppresentation of the corresponding Schur extensions of with generators g_{1}, ·.·, g_{n}, t_{1}, ·.·, t_{d}, c_{1}, ·.·c_{m} and relations

where the t_{i}'s commute modulo c_{1}, ·.·, c_{m} and the c_{i}'s are central.
SchurExtParPres(
G )
computes the Schur extensions corresponding to the ppowerpolypcpgroups G and returns them as ppowerpolypcpgroups.
SchurExtParPres(
ParPres ) F
computes a consistent pppresentation of Schur extensions of the groups defined by the record ParPres which describes ppowerpolypcpgroups. The output is a record rec(rel, expo, n, d, m, prime, cc, expo_vec, name), which describes the Schur extensions as ppowerpolypcpgroups; it is encoded in a form that it can be used as input for PPPPcpGroups.
gap> SchurExtParPres( ParPresGlobalVar_2_1[1] ); rec( prime := 2, rel := [ [ [ [ 7, 1 ] ] ], [ [ [ 2, 1 ], [ 3, 1+2*2^x ], [ 6, 12*2^x ] ], [ [ 3, 1 ], [ 5, 1 ] ] ], [ [ [ 3, 1+2*2^x ], [ 4, 1 ], [ 6, 22*2^x ] ], [ [ 3, 1 ] ], [ [ 4, 1 ], [ 6, 2*2^x ] ] ], [ [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 0 ] ] ], [ [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 0 ] ] ] , [ [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 0 ] ] ], [ [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 0 ] ] ] ], n := 2, d := 1, m := 4, expo := 2*2^x, expo_vec := [ 2, 0, 0, 0 ], cc := fail, name := "SchurExt_D" )
AbelianInvariantsMultiplier(
G ) F
computes the abelian invariants of the Schur multiplicators M(G) of the ppowerpolypcpgroups G. The output is a list [d_{1}, ·.·, d_{k}] consisting elements d_{i}, depending on the underlying parameter, such that M(G) ≅ C_{d1} ×…×C_{dk}.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < PPowerPolypcpgroups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> AbelianInvariantsMultiplier( G ); [ 2 ]
SchurMultiplicatorPPPPcps(
G ) F
computes the Schur multiplicators of the ppowerpolypcpgroups G and then returns the corresponding PPPPcpGroups.
gap> G := PPPPcpGroups( ParPresGlobalVar_3_1[1] ); < PPowerPoly pcpgroup with 5 generators of relative orders [ 3,3,3,3*3^x,3*3^x ] > gap> SchurMultiplicatorPPPPcps( G ); < PPowerPolypcpgroups with 2 generators of relative orders [ 3,9*3^x ] >
AbelianInvariants(
G ) F
computes the abelian invariants of the ppowerpolypcpgroups G and returns them as a list of list describing the parametrised elements.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < PPowerPolypcpgroups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> AbelianInvariants( G ); [ 2, 2 ]
ZeroCohomologyPPPPcps(
G[,
p] ) F
computes the zerothcohomology groups H^{0}(G,R) of the ppowerpolypcpgroups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a_{1},…, a_{k}] where the cohomology group is isomorphic to C_{a1} ×…×C_{ak} with C_{i} a cyclic group of order i (for i > 0) and C_{0} is interpreted as Z.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < PPowerPolypcpgroups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> ZeroCohomologyPPPPcp( G, 2 ); [ 2 ]
FirstCohomologyPPPPcps(
G[,
p] ) F
computes the firstcohomology groups H^{1}(G,R) of the ppowerpolypcpgroups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a_{1},…, a_{k}] where the cohomology group is isomorphic to C_{a1} ×…×C_{ak} with C_{i} a cyclic group of order i (for i > 0) and C_{0} is interpreted as Z.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < PPowerPolypcpgroups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> FirstCohomologyPPPPcps( G ); [ ]
SecondCohomologyPPPPcps(
G[,
p] ) F
computes the secondcohomology groups H^{2}(G,R) of the ppowerpolypcpgroups G with coefficients in R, where R ≅ GF(p) if the prime p is given or R ≅ Z otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a_{1},…, a_{k}] where the cohomology group is isomorphic to C_{a1} ×…×C_{ak} with C_{i} a cyclic group of order i (for i > 0) and C_{0} is interpreted as Z.
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] ); < PPowerPolypcpgroups with 3 generators of relative orders [ 2,2,2*2^x ] > gap> SecondCohomologyPPPPcps( G, 2 ); [ 2, 2, 2 ]
The following info classes are available
InfoConsistencyRelPPowerPoly V
level 1
the default value is 0.
InfoCollectingPPowerPoly V
level 1
the default value is 0.
SymbCompCC manual