Magma contains functions for constructing all metacyclic groups of order pn. It can also decide if a given p-group is metacyclic, construct invariants which distinguish this metacyclic group from all others of this order, and construct a standard presentation for the group.
This section describes the functions for accessing these algorithms. The functions were developed by Mike Newman, Eamonn O'Brien, and Michael Vaughan-Lee.
Return a list of the metacyclic groups of order pm, where p is a prime and n is a positive integer.PCGroups: BoolElt Default: trueIf true, the groups returned are in category GrpPC, otherwise they are in category GrpFP -- this will be faster if the groups have large class.
The group P is a p-group, either pc- or matrix or permutation group; if P is metacyclic, then return true, else false.
The group P is a metacyclic p-group, either pc- or matrix or permutation group; return tuple of invariants which uniquely identify metacyclic p-group P. This tuple which contains at least four terms, < r, s, t, n > has the following meaning: P has order pn + s; its derived quotient is Cpr x Cps; its derived group is cyclic of order pn - r; it has exponent pn + s - t.If p = 2, then additional invariants are needed to distinguish among the groups. We record the abelian invariants of the centre of P. If s = 1 and the centre of P has order 2, then the 2-group is maximal class and we record whether it is dihedral, quaternion or semidihedral. If s > 1 then the group has two cyclic central normal subgroups of order 2s - 1 whose central quotients are both semidihedral, or dihedral and quaternion. The invariant tuple has length at most 6.
The group P is a metacyclic p-group, either pc- or matrix or permutation group; return metacyclic p-group having a canonical pc-presentation which is isomorphic to P. If two metacyclic p-groups have the same canonical presentation, then they are isomorphic.
Return number of metacyclic groups of order pn.
Return true if for all primes p all p-quotients of the finitely-presented group G are metacyclic; otherwise return false and a description of the set of primes for which G has non-metacyclic p-quotient.If a prime p is supplied as a second argument, then the function returns true if all p-quotients of G are metacyclic; otherwise it returns false.
> X := MetacyclicPGroups (3, 6); > #X; 11 > X[4]; GrpPC of order 729 = 3^6 PC-Relations: $.1^3 = $.3, $.2^3 = $.4, $.3^3 = $.6, $.4^3 = $.5, $.5^3 = $.6, $.2^$.1 = $.2 * $.6^2 > H := SmallGroup (729, 59); > IsMetacyclicPGroup (H); true > I := InvariantsMetacyclicPGroup(H); > I; <2, 2, 2, 4, [], , > > S := StandardMetacyclicPGroup (H); GrpPC : S of order 729 = 3^6 PC-Relations: S.1^3 = S.3, S.2^3 = S.4, S.3^3 = S.6, S.4^3 = S.5, S.5^3 = S.6, S.2^S.1 = S.2 * S.6^2 > /* find this group in list */ > [IsIdenticalPresentation (S, X[i]): i in [1..#X]]; [ false, false, false, true, false, false, false, false, false, false, false ] > /* so this group is #4 in list */ > NumberOfMetacyclicPGroups (19, 7); 14 > Q := FreeGroup (4); > G := quo < Q | Q.2^2, Q.4^3, Q.2 * Q.3 * Q.2 * Q.3^-1, Q.1^9>; > /* are all p-quotients of G metacyclic? */ > HasAllPQuotientsMetacyclic (G); false [ 3 ] > /* the 3-quotient is not metacyclic */