> SetEchoInput(true);
> load "pcpres.m";
Loading "pcpres.m"
> /* setting up a finite polycyclic group -- Sym (4) */
> 
> G := PolycyclicGroup <x1, x2, x3, x4 | x1^2, x2^3 , x3^2, x4^2, x2^x1 =
>                x2^2, x3^x1 = x4, x3^x2 = x4, x4^x1 = x3, x4^x2 = x3 * x4 >;
> 
> G;
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G), 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.2^G.1 = G.2^2, 
    G.3^G.1 = G.4, 
    G.3^G.2 = G.4, 
    G.4^G.1 = G.3, 
    G.4^G.2 = G.3 * G.4
> 
> // does every element have unique normal form?
> IsConsistent (G);
true
> 
> #G;
24
> 
> flag, phi := IsIsomorphic (G, Sym (4));
> flag;
true
> phi (G.1);
(3, 4)
> phi (G.2);
(2, 4, 3)
> 
> // normal form? 
> g := G.3 * G.2^-1 * G.1^-1 * G.2 * G.4;
> g;
G.1 * G.2^2 * G.3 * G.4
> 
> // normal subgroups 
> N := NormalSubgroups (G);
> N;
Conjugacy classes of subgroups
------------------------------

[1]     Order 1            Length 1
        GrpPC of order 1
        PC-Relations:
[2]     Order 4            Length 1
        GrpPC of order 4 = 2^2
        PC-Relations:
            $.1^2 = Id($), 
            $.2^2 = Id($)
[3]     Order 12           Length 1
        GrpPC of order 12 = 2^2 * 3
        PC-Relations:
            $.1^3 = Id($), 
            $.2^2 = Id($), 
            $.3^2 = Id($), 
            $.2^$.1 = $.2 * $.3, 
            $.3^$.1 = $.2
[4]     Order 24           Length 1
        GrpPC of order 24 = 2^3 * 3
        PC-Relations:
            $.1^2 = Id($), 
            $.2^3 = Id($), 
            $.3^2 = Id($), 
            $.4^2 = Id($), 
            $.2^$.1 = $.2^2, 
            $.3^$.2 = $.3 * $.4, 
            $.4^$.1 = $.3 * $.4, 
            $.4^$.2 = $.3
> 
> // centraliser in G of g 
> C := Centraliser (G, g);
> 
> // subgroup generated by g 
> H := sub< G | g >;
> H;
GrpPC : H of order 4 = 2^2
PC-Relations:
    H.1^2 = H.2
> N := Normaliser (G, H);
> N;
GrpPC : N of order 8 = 2^3
PC-Relations:
    N.1^2 = N.3, 
    N.2^N.1 = N.2 * N.3
> 
> IdentifyGroup (N);
<8, 3>
> 
> /* set up infinite dihedral group */
> D<a,b> := PolycyclicGroup<a,b | a^2, b^a = b^-1>;
> D;
GrpGPC : D of infinite order on 2 PC-generators
PC-Relations:
    a^2 = Id(D), 
    b^a = b^-1
> 
> b^-2 * a * b^-1 * a^-1;
b^-1
> 
> /* exploring quotients of a finitely presented group */  
> 
> G<a,b> := Group <a, b |b^-2 * a * b * a * b * a,a^-1 * b * a^-1 * b * a^2 * 
b>;
> AbelianQuotientInvariants (G);
[ 3, 3 ]
> H, phi := pQuotient (G, 3, 0:Print:=1);
Class limit set to 127.

Lower exponent-3 central series for G

Group: G to lower exponent-3 central class 1 has order 3^2

Group: G to lower exponent-3 central class 2 has order 3^3

Group: G to lower exponent-3 central class 3 has order 3^5

Group completed. Lower exponent-3 central class = 3, Order = 3^5
> H;
GrpPC : H of order 243 = 3^5
PC-Relations:
    H.1^3 = H.4^2 * H.5, 
    H.2^3 = H.4^2 * H.5^2, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.5
> phi (a * b * a);
H.1^2 * H.2 * H.3
> IdentifyGroup (H);
<243, 7>
> 
> // aut gp of H 
> A := AutomorphismGroup (H);
> A;
A group of automorphisms of GrpPC : H
Generators:
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.2 * H.3 * H.5^2
        H.2 |--> H.1^2 * H.3 * H.5
        H.3 |--> H.3 * H.5
        H.4 |--> H.5
        H.5 |--> H.4^2
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.4^2
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.5
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1 * H.5^2
        H.2 |--> H.2
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.3
        H.3 |--> H.3 * H.4
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1 * H.3^2
        H.2 |--> H.2
        H.3 |--> H.3 * H.5
        H.4 |--> H.4
        H.5 |--> H.5
> #A;
2916
> 
> // specific aut 
> alpha := A.1 * A.2^-1 * A.3;
> alpha; 
Automorphism of GrpPC : H which maps:
    H.1 |--> H.2 * H.3 * H.4
    H.2 |--> H.1^2 * H.3 * H.5
    H.3 |--> H.3 * H.5
    H.4 |--> H.5
    H.5 |--> H.4^2
> 
> h := Random (H);
> h;
H.1^2 * H.2^2 * H.3^2
> alpha (h);
H.1 * H.2^2 * H.3^2 * H.4^2 * H.5
> 
> // perm rep for automorphism group: action on union of conjugacy classes 
> phi, B := ClassAction (A);
> B;
Permutation group B acting on a set of cardinality 108
Order = 2916 = 2^2 * 3^6
> CompositionFactors (B);
    G
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    1
> 
> // A is soluble so we can write down directly a pc-presentation for A
> X, tau := PCGroup (A);
> X;
GrpPC : X of order 2916 = 2^2 * 3^6
PC-Relations:
    X.1^2 = X.2, 
    X.2^2 = X.3 * X.4^2 * X.8^2, 
    X.3^3 = Id(X), 
    X.4^3 = Id(X), 
    X.5^3 = Id(X), 
    X.6^3 = Id(X), 
    X.7^3 = Id(X), 
    X.8^3 = Id(X), 
    X.3^X.1 = X.3 * X.4, 
    X.4^X.1 = X.4^2, 
    X.5^X.1 = X.5^2 * X.8^2, 
    X.6^X.1 = X.6 * X.7 * X.8^2, 
    X.6^X.2 = X.6^2, 
    X.7^X.1 = X.6 * X.7^2 * X.8, 
    X.7^X.2 = X.7^2, 
    X.7^X.6 = X.7 * X.8
> Y := AutomorphismGroup (X);
> #Y;
373248
> 
> U := PCGroup (Y);
> U;
GrpPC : U of order 373248 = 2^9 * 3^6
PC-Relations:
    U.1^2 = Id(U), 
    U.2^2 = U.9 * U.10 * U.14^2, 
    U.3^2 = U.4, 
    U.4^2 = U.5, 
    U.5^2 = Id(U), 
    U.6^3 = Id(U), 
    U.7^2 = Id(U), 
    U.8^3 = Id(U), 
    U.9^2 = U.11, 
    U.10^2 = U.11, 
    U.11^2 = Id(U), 
    U.12^3 = Id(U), 
    U.13^3 = Id(U), 
    U.14^3 = Id(U), 
    U.15^3 = Id(U), 
    U.3^U.2 = U.3 * U.4, 
    U.4^U.2 = U.4 * U.5, 
    U.6^U.1 = U.6^2, 
    U.6^U.2 = U.6^2, 
    U.6^U.3 = U.6^2, 
    U.7^U.2 = U.7 * U.8^2 * U.9 * U.14^2 * U.15^2, 
    U.8^U.2 = U.8^2 * U.9 * U.11 * U.15, 
    U.8^U.7 = U.8^2, 
    U.9^U.2 = U.10 * U.15^2, 
    U.9^U.7 = U.9 * U.10 * U.11, 
    U.9^U.8 = U.9 * U.10, 
    U.10^U.2 = U.9 * U.11 * U.14, 
    U.10^U.7 = U.10 * U.11, 
    U.10^U.8 = U.9 * U.11, 
    U.10^U.9 = U.10 * U.11, 
    U.11^U.2 = U.11 * U.14^2 * U.15^2, 
    U.12^U.2 = U.13, 
    U.12^U.3 = U.12^2 * U.13^2, 
    U.12^U.4 = U.13^2, 
    U.12^U.5 = U.12^2, 
    U.13^U.2 = U.12, 
    U.13^U.3 = U.12 * U.13^2, 
    U.13^U.4 = U.12, 
    U.13^U.5 = U.13^2, 
    U.14^U.2 = U.15, 
    U.14^U.9 = U.14^2 * U.15, 
    U.14^U.10 = U.15^2, 
    U.14^U.11 = U.14^2, 
    U.15^U.2 = U.14 * U.15, 
    U.15^U.7 = U.15^2, 
    U.15^U.8 = U.14^2 * U.15, 
    U.15^U.9 = U.14 * U.15, 
    U.15^U.10 = U.14, 
    U.15^U.11 = U.15^2
> 
> // family of 7-groups 
> F<a, b> := FreeGroup (2);
>  p := 7;
>  Q := [];
>  for k := 1 to p - 1 do
>        G := quo< F | a^p = (b, a, a), b^p = a^(k*p), (b, a, b)>;
>        Q[k] := pQuotient (G, p, 10);
>  end for;
> 
> Q;
[
    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^2, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^3, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^5, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^6, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4
]
> 
> // how many distinct isom types?
> S := [StandardPresentation (H): H in Q];
> 
> // generic function to identify different objects in S
> // using MyCompare 
> FindDifferent := function (S, MyCompare)
> 
>    if #S le 1 then return S, [i : i in [1..#S]]; end if;
>    D := [S[1]];
>    for i in [2..#S] do
>      if forall {j : j in [1..#D] | 
>                  MyCompare (S[i], D[j]) eq false}  then 
>         Append (~D, S[i]);
>      end if;
>    end for;
>    pos := [Position (S, D[i]) : i in [1..#D]];
>    return D, pos;
> end function;
> 
> T, pos := FindDifferent (S, IsIdenticalPresentation);
> // non-isomorphic groups 
> T;
[
    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.2^7 = $.4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.2^7 = $.4^3, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4
]
> 
> // see Cavicchioli, O'Brien and Spaggiari, J. Algebra 2008 
> // www.math.auckland.ac.nz/~obrien/research/fib.pdf
> // generalised Fibonacci type groups 
> 
> CHR := function (n, m, k)
>    F := FreeGroup (n);
>    R := [];
>    for i in [1..n] do 
>       a := i + m;
>       if a gt n then repeat a := a - n; until a le n; end if;
>       b := i + k;
>       if b gt n then repeat b := b - n; until b le n; end if;
>       Append (~R, F.i * F.a = F.b);
>    end for;
>    Q := quo <F | R>;
>    return Q;
> end function;
> 
> /* prove that 9, 3, 4 is infinite */
> G := CHR (9, 3, 4);
> D := DerivedGroup (G);
> S := DerivedGroup (D);
> R := Rewrite (D, S: Simplify:=false);
> b := Ngens (R);
> r := #Relations (R);
> "# of generators, # relations ", b, r;
# of generators, # relations  321 768
> 
> // Golod-Safarevic theorem 
> // G p-group has Frattini quotient rank d and r relators; 
> // if r < d^2/4 then G is infinite 
> 
> // stronger version of this theorem 
> Left := func<b, r | r - b >;
> 
> Right := func<d, e, p | d^2 div 2  + (-1)^p * (d div 2) - d - e
>             + (e - (-1)^p * (d div 2) - (d^2 div 4)) * (d div 2)>;
> 
> P:= pQuotient(R, 2, 2:Print:=1);

Lower exponent-2 central series for R

Group: R to lower exponent-2 central class 1 has order 2^43

Group: R to lower exponent-2 central class 2 has order 2^604
> d := FrattiniQuotientRank (P); 
> // rank of class 2 section 
> e := NPCgens (P) - d;
> p := 2;
> 
> left := Left (b, r); left;
447
> right := Right (d, e, p); right;
1979
> left le right;
true
> // so G is infinite
> 
> // an unsolved case 
> G := CHR (9, 1, 3);
> G;
Finitely presented group G on 9 generators
Relations
    G.1 * G.2 = G.4
    G.2 * G.3 = G.5
    G.3 * G.4 = G.6
    G.4 * G.5 = G.7
    G.5 * G.6 = G.8
    G.6 * G.7 = G.9
    G.7 * G.8 = G.1
    G.8 * G.9 = G.2
    G.9 * G.1 = G.3
> 
> // G/G'
> AbelianQuotientInvariants (G);
[ 19 ]
> 
> // Reduce # of generators 
> S := Simplify (G);
> S;
Finitely presented group S on 2 generators
Generators as words in group G
    S.1 = G.2
    S.2 = G.5
Relations
    S.2^-2 * S.1^-2 * S.2^-1 * S.1^2 * S.2 * S.1 * S.2^-1 * S.1 = Id(S)
    S.2^-1 * S.1^-1 * S.2^-2 * S.1^-1 * S.2^-1 * S.1 * S.2^-2 * S.1^-1 * S.2 * 
    S.1^-1 = Id(S)
> P := Homomorphisms (S, PSL(2, 8));
> P;
[
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 5, 7, 4, 9, 6, 8, 3, 2)
        S.2 |--> (1, 6, 3, 5, 9, 8, 7, 2, 4),
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 7, 9, 8, 2, 5, 4, 6, 3)
        S.2 |--> (1, 8, 4, 3, 2, 7, 5, 9, 6),
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 9, 2, 4, 3, 7, 8, 5, 6)
        S.2 |--> (1, 8, 2, 3, 5, 9, 4, 6, 7)
]
> 
> // easy to get homomorphism from G to PSL(2, 8) x Z_19
> // Not too difficult to show kernel K is perfect 
> // Hard question: is K infinite simple, trivial, finite?
> 
> // Baumslag-Solitar groups
> 
> G<a, b> := Group<a, b | a*b^3*a^-1 = b^5>;
> AbelianQuotientInvariants (G);
[ 2, 0 ]
> 
> // p-quotient 
> H, tau := pQuotient (G, 2, 5:Print:=1);

Lower exponent-2 central series for G

Group: G to lower exponent-2 central class 1 has order 2^2

Group: G to lower exponent-2 central class 2 has order 2^4

Group: G to lower exponent-2 central class 3 has order 2^6

Group: G to lower exponent-2 central class 4 has order 2^8

Group: G to lower exponent-2 central class 5 has order 2^10
> H;
GrpPC : H of order 1024 = 2^10
PC-Relations:
    H.1^2 = H.4, 
    H.2^2 = H.3 * H.5 * H.7, 
    H.3^2 = H.5 * H.7 * H.9, 
    H.4^2 = H.6, 
    H.5^2 = H.7 * H.9, 
    H.6^2 = H.8, 
    H.7^2 = H.9, 
    H.8^2 = H.10, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.5, 
    H.4^H.2 = H.4 * H.9, 
    H.5^H.1 = H.5 * H.7, 
    H.7^H.1 = H.7 * H.9
> tau;
Mapping from: GrpFP: G to GrpPC: H
> (a * b^2 *a) @ tau;
H.3 * H.4 * H.7 * H.9
> 
> // nilpotent quotient 
> K, phi := NilpotentQuotient (G, 5);
> K;
GrpGPC : K of infinite order on 6 PC-generators
PC-Relations:
    K.2^2 = K.3 * K.4 * K.5, 
    K.3^2 = K.4 * K.5 * K.6, 
    K.4^2 = K.5 * K.6, 
    K.5^2 = K.6, 
    K.6^2 = Id(K), 
    K.2^K.1 = K.2 * K.3, 
    K.2^(K.1^-1) = K.2 * K.3 * K.6, 
    K.3^K.1 = K.3 * K.4, 
    K.3^(K.1^-1) = K.3 * K.4, 
    K.4^K.1 = K.4 * K.5, 
    K.4^(K.1^-1) = K.4 * K.5, 
    K.5^K.1 = K.5 * K.6, 
    K.5^(K.1^-1) = K.5 * K.6
> phi;
Mapping from: GrpFP: G to GrpGPC: K
> K.5 @@ phi;
(b, a, a, a)
> 
> // soluble group use PCGroup to write down PCPresentation 
> G := WreathProduct (Alt(4), DihedralGroup(10));
> IsSoluble (G); 
true
> P, pi := PCGroup (G);
> 
> // soluble quotient
> G := WreathProduct(Sym(5), DihedralGroup(3));
> #G;
10368000
> F := FPGroupStrong (G);
> time S, phi := SolubleQuotient (F);
Time: 0.190
> S;
GrpPC : S of order 48 = 2^4 * 3
PC-Relations:
    S.1^2 = Id(S), 
    S.2^2 = Id(S), 
    S.3^3 = Id(S), 
    S.4^2 = Id(S), 
    S.5^2 = Id(S), 
    S.3^S.2 = S.3^2, 
    S.4^S.3 = S.5, 
    S.5^S.2 = S.4 * S.5, 
    S.5^S.3 = S.4 * S.5
> 
> // find p-groups G where G^p \leq Z(G)
> //is p-group pcentral?
> IsPCentral := function(G)
>    Z := Center (G);
>    O := Omega (G, 1);
>    return O subset Z;
> end function;
> 
> // groups of order 32 with pcentral property
> X := SmallGroups (32, IsPCentral);
> "Number of examples is ", #X;
Number of examples is  20
> X[1];
GrpPC of order 32 = 2^5
PC-Relations:
    $.1^2 = $.2, 
    $.2^2 = $.3, 
    $.3^2 = $.4, 
    $.4^2 = $.5
> 
> // identify abelian invariants of Schur multiplier of G 
> SchurMultiplier := function (G)
>    B := PrimeBasis (#G);
>    M := [pMultiplicator (G, p): p in B];
>    M := [x : x in M | #x gt 0 and x ne [1]];
>    return M;
> end function;
> 
> // extension of soluble group 
> D4 := DihedralGroup (4);
> SchurMultiplier (D4);
[
    [ 2 ]
]
> 
> // largest central extension 
> D := Darstellungsgruppe (FPGroup (D4));
> D;
Finitely presented group D on 3 generators
Relations
    D.2^2 = Id(D)
    D.1^4 * D.3 = Id(D)
    (D.1^-1 * D.2)^2 = Id(D)
    (D.1, D.3) = Id(D)
    (D.2, D.3) = Id(D)
> H := pQuotient (D, 2, 10:Print := 1);

Lower exponent-2 central series for D

Group: D to lower exponent-2 central class 1 has order 2^2

Group: D to lower exponent-2 central class 2 has order 2^3

Group: D to lower exponent-2 central class 3 has order 2^4

Group completed. Lower exponent-2 central class = 3, Order = 2^4
> H;
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.1^2 = H.3, 
    H.3^2 = H.4, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.2 = H.3 * H.4
> 
> // extensions of D4 by C2 
> S := ExtensionsOfSolubleGroup (D4, CyclicGroup (2));
> X := [pQuotient(x, 2, 10:Print:=0): x in S];
> [IdentifyGroup (x): x in X];
[ <16, 11>, <16, 13>, <16, 7>, <16, 8> ]
> 
> quit;

Total time: 5.480 seconds, Total memory usage: 47.19MB
> 
> /* setting up a finite polycyclic group -- Sym (4) */
> 
> G := PolycyclicGroup <x1, x2, x3, x4 | x1^2, x2^3 , x3^2, x4^2, x2^x1 =
>                x2^2, x3^x1 = x4, x3^x2 = x4, x4^x1 = x3, x4^x2 = x3 * x4 >;
> 
> G;
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G), 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.2^G.1 = G.2^2, 
    G.3^G.1 = G.4, 
    G.3^G.2 = G.4, 
    G.4^G.1 = G.3, 
    G.4^G.2 = G.3 * G.4
> 
> // does every element have unique normal form?
> IsConsistent (G);
true
> 
> #G;
24
> 
> flag, phi := IsIsomorphic (G, Sym (4));
> flag;
true
> phi (G.1);
(3, 4)
> phi (G.2);
(2, 4, 3)
> 
> // normal form? 
> g := G.3 * G.2^-1 * G.1^-1 * G.2 * G.4;
> g;
G.1 * G.2^2 * G.3 * G.4
> 
> // normal subgroups 
> N := NormalSubgroups (G);
> N;
Conjugacy classes of subgroups
------------------------------

[1]     Order 1            Length 1
        GrpPC of order 1
        PC-Relations:
[2]     Order 4            Length 1
        GrpPC of order 4 = 2^2
        PC-Relations:
            $.1^2 = Id($), 
            $.2^2 = Id($)
[3]     Order 12           Length 1
        GrpPC of order 12 = 2^2 * 3
        PC-Relations:
            $.1^3 = Id($), 
            $.2^2 = Id($), 
            $.3^2 = Id($), 
            $.2^$.1 = $.2 * $.3, 
            $.3^$.1 = $.2
[4]     Order 24           Length 1
        GrpPC of order 24 = 2^3 * 3
        PC-Relations:
            $.1^2 = Id($), 
            $.2^3 = Id($), 
            $.3^2 = Id($), 
            $.4^2 = Id($), 
            $.2^$.1 = $.2^2, 
            $.3^$.2 = $.3 * $.4, 
            $.4^$.1 = $.3 * $.4, 
            $.4^$.2 = $.3
> 
> // centraliser in G of g 
> C := Centraliser (G, g);
> 
> // subgroup generated by g 
> H := sub< G | g >;
> H;
GrpPC : H of order 4 = 2^2
PC-Relations:
    H.1^2 = H.2
> N := Normaliser (G, H);
> N;
GrpPC : N of order 8 = 2^3
PC-Relations:
    N.1^2 = N.3, 
    N.2^N.1 = N.2 * N.3
> 
> IdentifyGroup (N);
<8, 3>
> 
> /* set up infinite dihedral group */
> D<a,b> := PolycyclicGroup<a,b | a^2, b^a = b^-1>;
> D;
GrpGPC : D of infinite order on 2 PC-generators
PC-Relations:
    a^2 = Id(D), 
    b^a = b^-1
> 
> b^-2 * a * b^-1 * a^-1;
b^-1
> 
> /* exploring quotients of a finitely presented group */  
> 
> G<a,b> := Group <a, b |b^-2 * a * b * a * b * a,a^-1 * b * a^-1 * b * a^2 * 
b>;
> AbelianQuotientInvariants (G);
[ 3, 3 ]
> H, phi := pQuotient (G, 3, 0:Print:=1);
Class limit set to 127.

Lower exponent-3 central series for G

Group: G to lower exponent-3 central class 1 has order 3^2

Group: G to lower exponent-3 central class 2 has order 3^3

Group: G to lower exponent-3 central class 3 has order 3^5

Group completed. Lower exponent-3 central class = 3, Order = 3^5
> H;
GrpPC : H of order 243 = 3^5
PC-Relations:
    H.1^3 = H.4^2 * H.5, 
    H.2^3 = H.4^2 * H.5^2, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.5
> phi (a * b * a);
H.1^2 * H.2 * H.3
> IdentifyGroup (H);
<243, 7>
> 
> // aut gp of H 
> A := AutomorphismGroup (H);
> A;
A group of automorphisms of GrpPC : H
Generators:
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.2 * H.3 * H.5^2
        H.2 |--> H.1^2 * H.3 * H.5
        H.3 |--> H.3 * H.5
        H.4 |--> H.5
        H.5 |--> H.4^2
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.4^2
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.5
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1 * H.5^2
        H.2 |--> H.2
        H.3 |--> H.3
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1
        H.2 |--> H.2 * H.3
        H.3 |--> H.3 * H.4
        H.4 |--> H.4
        H.5 |--> H.5
    Automorphism of GrpPC : H which maps:
        H.1 |--> H.1 * H.3^2
        H.2 |--> H.2
        H.3 |--> H.3 * H.5
        H.4 |--> H.4
        H.5 |--> H.5
> #A;
2916
> 
> // specific aut 
> alpha := A.1 * A.2^-1 * A.3;
> alpha; 
Automorphism of GrpPC : H which maps:
    H.1 |--> H.2 * H.3 * H.4
    H.2 |--> H.1^2 * H.3 * H.5
    H.3 |--> H.3 * H.5
    H.4 |--> H.5
    H.5 |--> H.4^2
> 
> h := Random (H);
> h;
H.2 * H.4 * H.5
> alpha (h);
H.1^2 * H.3 * H.4^2 * H.5^2
> 
> // perm rep for automorphism group: action on union of conjugacy classes 
> phi, B := ClassAction (A);
> B;
Permutation group B acting on a set of cardinality 108
Order = 2916 = 2^2 * 3^6
> CompositionFactors (B);
    G
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    1
> 
> // A is soluble so we can write down directly a pc-presentation for A
> X, tau := PCGroup (A);
> X;
GrpPC : X of order 2916 = 2^2 * 3^6
PC-Relations:
    X.1^2 = X.2, 
    X.2^2 = X.3 * X.4^2 * X.8^2, 
    X.3^3 = Id(X), 
    X.4^3 = Id(X), 
    X.5^3 = Id(X), 
    X.6^3 = Id(X), 
    X.7^3 = Id(X), 
    X.8^3 = Id(X), 
    X.3^X.1 = X.3 * X.4, 
    X.4^X.1 = X.4^2, 
    X.5^X.1 = X.5^2 * X.8^2, 
    X.6^X.1 = X.6 * X.7 * X.8^2, 
    X.6^X.2 = X.6^2, 
    X.7^X.1 = X.6 * X.7^2 * X.8, 
    X.7^X.2 = X.7^2, 
    X.7^X.6 = X.7 * X.8
> Y := AutomorphismGroup (X);
> #Y;
373248
> 
> U := PCGroup (Y);
> U;
GrpPC : U of order 373248 = 2^9 * 3^6
PC-Relations:
    U.1^2 = Id(U), 
    U.2^2 = U.5 * U.12, 
    U.3^2 = U.4, 
    U.4^2 = U.5, 
    U.5^2 = Id(U), 
    U.6^3 = Id(U), 
    U.7^2 = Id(U), 
    U.8^3 = Id(U), 
    U.9^2 = U.11, 
    U.10^2 = U.11, 
    U.11^2 = Id(U), 
    U.12^3 = Id(U), 
    U.13^3 = Id(U), 
    U.14^3 = Id(U), 
    U.15^3 = Id(U), 
    U.3^U.2 = U.3 * U.4 * U.13^2, 
    U.4^U.2 = U.4 * U.5 * U.12 * U.13, 
    U.5^U.2 = U.5 * U.12 * U.13^2, 
    U.6^U.1 = U.6^2, 
    U.6^U.3 = U.6^2, 
    U.8^U.7 = U.8^2, 
    U.9^U.7 = U.9 * U.11, 
    U.9^U.8 = U.10 * U.11, 
    U.10^U.7 = U.9 * U.10, 
    U.10^U.8 = U.9 * U.10 * U.11, 
    U.10^U.9 = U.10 * U.11, 
    U.12^U.2 = U.13, 
    U.12^U.3 = U.12 * U.13, 
    U.12^U.4 = U.12^2 * U.13, 
    U.12^U.5 = U.12^2, 
    U.13^U.2 = U.12^2, 
    U.13^U.3 = U.12, 
    U.13^U.4 = U.12 * U.13, 
    U.13^U.5 = U.13^2, 
    U.14^U.7 = U.14 * U.15, 
    U.14^U.8 = U.15, 
    U.14^U.9 = U.14 * U.15, 
    U.14^U.10 = U.15^2, 
    U.14^U.11 = U.14^2, 
    U.15^U.7 = U.15^2, 
    U.15^U.8 = U.14^2 * U.15^2, 
    U.15^U.9 = U.14 * U.15^2, 
    U.15^U.10 = U.14, 
    U.15^U.11 = U.15^2
> 
> // family of 7-groups 
> F<a, b> := FreeGroup (2);
>  p := 7;
>  Q := [];
>  for k := 1 to p - 1 do
>        G := quo< F | a^p = (b, a, a), b^p = a^(k*p), (b, a, b)>;
>        Q[k] := pQuotient (G, p, 10);
>  end for;
> 
> Q;
[
    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^2, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^3, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^5, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.1^7 = $.4, 
        $.2^7 = $.4^6, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4
]
> 
> // how many distinct isom types?
> S := [StandardPresentation (H): H in Q];
> 
> // generic function to identify different objects in S
> // using MyCompare 
> FindDifferent := function (S, MyCompare)
> 
>    if #S le 1 then return S, [i : i in [1..#S]]; end if;
>    D := [S[1]];
>    for i in [2..#S] do
>      if forall {j : j in [1..#D] | 
>                  MyCompare (S[i], D[j]) eq false}  then 
>         Append (~D, S[i]);
>      end if;
>    end for;
>    pos := [Position (S, D[i]) : i in [1..#D]];
>    return D, pos;
> end function;
> 
> T, pos := FindDifferent (S, IsIdenticalPresentation);
> // non-isomorphic groups 
> T;
[
    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.2^7 = $.4, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4,

    GrpPC of order 2401 = 7^4
    PC-Relations:
        $.2^7 = $.4^3, 
        $.2^$.1 = $.2 * $.3, 
        $.3^$.1 = $.3 * $.4
]
> 
> // see Cavicchioli, O'Brien and Spaggiari, J. Algebra 2008 
> // www.math.auckland.ac.nz/~obrien/research/fib.pdf
> // generalised Fibonacci type groups 
> 
> CHR := function (n, m, k)
>    F := FreeGroup (n);
>    R := [];
>    for i in [1..n] do 
>       a := i + m;
>       if a gt n then repeat a := a - n; until a le n; end if;
>       b := i + k;
>       if b gt n then repeat b := b - n; until b le n; end if;
>       Append (~R, F.i * F.a = F.b);
>    end for;
>    Q := quo <F | R>;
>    return Q;
> end function;
> 
> /* prove that 9, 3, 4 is infinite */
> G := CHR (9, 3, 4);
> D := DerivedGroup (G);
> S := DerivedGroup (D);
> R := Rewrite (D, S: Simplify:=false);
> b := Ngens (R);
> r := #Relations (R);
> "# of generators, # relations ", b, r;
# of generators, # relations  321 768
> 
> // Golod-Safarevic theorem 
> // G p-group has Frattini quotient rank d and r relators; 
> // if r < d^2/4 then G is infinite 
> 
> // stronger version of this theorem 
> Left := func<b, r | r - b >;
> 
> Right := func<d, e, p | d^2 div 2  + (-1)^p * (d div 2) - d - e
>             + (e - (-1)^p * (d div 2) - (d^2 div 4)) * (d div 2)>;
> 
> P:= pQuotient(R, 2, 2:Print:=1);

Lower exponent-2 central series for R

Group: R to lower exponent-2 central class 1 has order 2^43

Group: R to lower exponent-2 central class 2 has order 2^604
> d := FrattiniQuotientRank (P); 
> // rank of class 2 section 
> e := NPCgens (P) - d;
> p := 2;
> 
> left := Left (b, r); left;
447
> right := Right (d, e, p); right;
1979
> left le right;
true
> // so G is infinite
> 
> // an unsolved case 
> G := CHR (9, 1, 3);
> G;
Finitely presented group G on 9 generators
Relations
    G.1 * G.2 = G.4
    G.2 * G.3 = G.5
    G.3 * G.4 = G.6
    G.4 * G.5 = G.7
    G.5 * G.6 = G.8
    G.6 * G.7 = G.9
    G.7 * G.8 = G.1
    G.8 * G.9 = G.2
    G.9 * G.1 = G.3
> 
> // G/G'
> AbelianQuotientInvariants (G);
[ 19 ]
> 
> // Reduce # of generators 
> S := Simplify (G);
> S;
Finitely presented group S on 2 generators
Generators as words in group G
    S.1 = G.2
    S.2 = G.5
Relations
    S.2^-2 * S.1^-2 * S.2^-1 * S.1^2 * S.2 * S.1 * S.2^-1 * S.1 = Id(S)
    S.2^-1 * S.1^-1 * S.2^-2 * S.1^-1 * S.2^-1 * S.1 * S.2^-2 * S.1^-1 * S.2 * 
    S.1^-1 = Id(S)
> P := Homomorphisms (S, PSL(2, 8));
> P;
[
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 5, 7, 4, 9, 6, 8, 3, 2)
        S.2 |--> (1, 6, 3, 5, 9, 8, 7, 2, 4),
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 7, 9, 8, 2, 5, 4, 6, 3)
        S.2 |--> (1, 8, 4, 3, 2, 7, 5, 9, 6),
    Homomorphism of GrpFP: S into GrpPerm: $, Degree 9, Order 2^3 * 3^2 * 7 
    induced by
        S.1 |--> (1, 9, 2, 4, 3, 7, 8, 5, 6)
        S.2 |--> (1, 8, 2, 3, 5, 9, 4, 6, 7)
]
> 
> // easy to get homomorphism from G to PSL(2, 8) x Z_19
> // Not too difficult to show kernel K is perfect 
> // Hard question: is K infinite simple, trivial, finite?
> 
> // Baumslag-Solitar groups
> 
> G<a, b> := Group<a, b | a*b^3*a^-1 = b^5>;
> AbelianQuotientInvariants (G);
[ 2, 0 ]
> 
> // p-quotient 
> H, tau := pQuotient (G, 2, 5:Print:=1);

Lower exponent-2 central series for G

Group: G to lower exponent-2 central class 1 has order 2^2

Group: G to lower exponent-2 central class 2 has order 2^4

Group: G to lower exponent-2 central class 3 has order 2^6

Group: G to lower exponent-2 central class 4 has order 2^8

Group: G to lower exponent-2 central class 5 has order 2^10
> H;
GrpPC : H of order 1024 = 2^10
PC-Relations:
    H.1^2 = H.4, 
    H.2^2 = H.3 * H.5 * H.7, 
    H.3^2 = H.5 * H.7 * H.9, 
    H.4^2 = H.6, 
    H.5^2 = H.7 * H.9, 
    H.6^2 = H.8, 
    H.7^2 = H.9, 
    H.8^2 = H.10, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.5, 
    H.4^H.2 = H.4 * H.9, 
    H.5^H.1 = H.5 * H.7, 
    H.7^H.1 = H.7 * H.9
> tau;
Mapping from: GrpFP: G to GrpPC: H
> (a * b^2 *a) @ tau;
H.3 * H.4 * H.7 * H.9
> 
> // nilpotent quotient 
> K, phi := NilpotentQuotient (G, 5);
> K;
GrpGPC : K of infinite order on 6 PC-generators
PC-Relations:
    K.2^2 = K.3 * K.4 * K.5, 
    K.3^2 = K.4 * K.5 * K.6, 
    K.4^2 = K.5 * K.6, 
    K.5^2 = K.6, 
    K.6^2 = Id(K), 
    K.2^K.1 = K.2 * K.3, 
    K.2^(K.1^-1) = K.2 * K.3 * K.6, 
    K.3^K.1 = K.3 * K.4, 
    K.3^(K.1^-1) = K.3 * K.4, 
    K.4^K.1 = K.4 * K.5, 
    K.4^(K.1^-1) = K.4 * K.5, 
    K.5^K.1 = K.5 * K.6, 
    K.5^(K.1^-1) = K.5 * K.6
> phi;
Mapping from: GrpFP: G to GrpGPC: K
> K.5 @@ phi;
(b, a, a, a)
> 
> // soluble group use PCGroup to write down PCPresentation 
> G := WreathProduct (Alt(4), DihedralGroup(10));
> IsSoluble (G); 
true
> P, pi := PCGroup (G);
> 
> // soluble quotient
> G := WreathProduct(Sym(5), DihedralGroup(3));
> #G;
10368000
> F := FPGroupStrong (G);
> time S, phi := SolubleQuotient (F);
Time: 0.210
> S;
GrpPC : S of order 48 = 2^4 * 3
PC-Relations:
    S.1^2 = Id(S), 
    S.2^2 = Id(S), 
    S.3^3 = Id(S), 
    S.4^2 = Id(S), 
    S.5^2 = Id(S), 
    S.3^S.2 = S.3^2, 
    S.4^S.3 = S.5, 
    S.5^S.2 = S.4 * S.5, 
    S.5^S.3 = S.4 * S.5
> 
> // find p-groups G where G^p \leq Z(G)
> //is p-group pcentral?
> IsPCentral := function(G)
>    Z := Center (G);
>    O := Omega (G, 1);
>    return O subset Z;
> end function;
> 
> // groups of order 32 with pcentral property
> X := SmallGroups (32, IsPCentral);
> "Number of examples is ", #X;
Number of examples is  20
> X[1];
GrpPC of order 32 = 2^5
PC-Relations:
    $.1^2 = $.2, 
    $.2^2 = $.3, 
    $.3^2 = $.4, 
    $.4^2 = $.5
> 
> // identify abelian invariants of Schur multiplier of G 
> SchurMultiplier := function (G)
>    B := PrimeBasis (#G);
>    M := [pMultiplicator (G, p): p in B];
>    M := [x : x in M | #x gt 0 and x ne [1]];
>    return M;
> end function;
> 
> // extension of soluble group 
> D4 := DihedralGroup (4);
> SchurMultiplier (D4);
[
    [ 2 ]
]
> 
> // largest central extension 
> D := Darstellungsgruppe (FPGroup (D4));
> D;
Finitely presented group D on 3 generators
Relations
    D.2^2 = Id(D)
    D.1^4 * D.3 = Id(D)
    (D.1^-1 * D.2)^2 = Id(D)
    (D.1, D.3) = Id(D)
    (D.2, D.3) = Id(D)
> H := pQuotient (D, 2, 10:Print := 1);

Lower exponent-2 central series for D

Group: D to lower exponent-2 central class 1 has order 2^2

Group: D to lower exponent-2 central class 2 has order 2^3

Group: D to lower exponent-2 central class 3 has order 2^4

Group completed. Lower exponent-2 central class = 3, Order = 2^4
> H;
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.1^2 = H.3, 
    H.3^2 = H.4, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.2 = H.3 * H.4
> 
> // extensions of D4 by C2 
> S := ExtensionsOfSolubleGroup (D4, CyclicGroup (2));
> X := [pQuotient(x, 2, 10:Print:=0): x in S];
> [IdentifyGroup (x): x in X];
[ <16, 11>, <16, 13>, <16, 7>, <16, 8> ]
> 
> quit;

Total time: 5.620 seconds, Total memory usage: 47.19MB