Database of Transitive Groups#

Magma has a database containing all transitive permutation groups, up to conjugacy in the ambient symmetric group, having degree up to 48, and one containing all primitive permutation groups with degree less than 4096.

The transitive groups up to degree 15 were determined by Greg Butler and John McKay, the groups having degree in the range 16 to 30 were determined by Alexander Hulpke [Hulpke, 2005]. John Cannon and Derek Holt [Cannon and Holt, 2008] have determined the transitive groups of degree 32. Derek Holt and Gordon Royle have determined the transitive groups with degrees 33 to 47 [Holt and Royle, 2019], and Derek Holt has determined the transitive groups of degree 48.

Accessing the Databases#

TransitiveGroupDatabaseLimit() -> RngIntElt#: The limiting degree of the database of transitive groups.

NumberOfTransitiveGroups(d): RngIntElt -> RngIntElt#: Given a degree \(d\) in the required range, return the number of transitive groups of degree \(d\).

TransitiveGroup(d, n): RngIntElt, RngIntElt -> GrpPerm, MonStgElt#: Given a degree \(d\) in the required range and a positive integer \(n\), return the \(n\)-th transitive group of degree \(d\). Also returns a string giving a description of the group.

TransitiveGroupDescription(d, n): RngIntElt, RngIntElt -> MonStgElt#: A string giving a description of the \(n\)-th transitive group of degree \(d\).

TransitiveGroupDescription(G): GrpPerm -> MonStgElt#: A string giving a description of the transitive group \(G\).

TransitiveGroup(d): RngIntElt -> GrpPerm, MonStgElt#: Given a degree \(d\) in the required range, return the first transitive group of degree \(d\). Also returns a string giving a description of the group.

TransitiveGroup(d, f): RngIntElt, Program -> GrpPerm, MonStgElt#: Given a degree \(d\) in the required range and a predicate \(f\) (as a function or intrinsic), return the first transitive group of degree \(d\) which satisfies \(f\). Also returns a string giving a description of the group.

TransitiveGroup(S, f): [RngIntElt], Program -> GrpPerm, MonStgElt#: Given a sequence \(S\) of degrees and a predicate \(f\) (as a function or intrinsic), return the first transitive group with degree in \(S\) which satisfies \(f\). Also returns a string giving a description of the group.

TransitiveGroups(d: parameters): RngIntElt -> [GrpPerm]#

Warning: BoolElt                    Default: true

Return a sequence of all transitive groups of degree \(d\) up to conjugacy in the symmetric group \(S_d\). Some degrees will produce a very large sequence of groups – in such cases a warning will be printed unless the user specifies Warning := false.

TransitiveGroups(S: parameters): [RngIntElt] -> [GrpPerm]#

Warning: BoolElt                    Default: true

Given a sequence \(S\) of degrees, return a sequence of all transitive groups with degree in \(S\). The resulting sequence may be very long – in such cases a warning will be printed unless the user specifies Warning := false.

TransitiveGroups(d, f): RngIntElt, Program -> [GrpPerm]#: Given an integer \(d\) and a predicate (function or intrinsic) \(f\), return a sequence containing all transitive groups \(G\) of degree \(d\) satisfying f(G) eq true.

TransitiveGroups(S, f): [RngIntElt], Program -> [GrpPerm]#: Given a sequence \(S\) of degrees and a predicate (function or intrinsic) \(f\), return a sequence containing all transitive groups \(G\) with degree in \(S\) satisfying f(G) eq true.

Example: Transitive (ex-95bf5a)#

We apply some of these functions to the degree \(8\) case.

> NumberOfTransitiveGroups(8);
50
> TransitiveGroup(8, 3);
Permutation group acting on a set of cardinality 8
    (1, 2)(3, 4)(5, 6)(7, 8)
    (1, 4)(2, 3)(5, 8)(6, 7)
    (1, 8)(2, 7)(3, 6)(4, 5)
E(8) = 2[x]2[x]2
> S := TransitiveGroups(8, IsPrimitive);
> #S;
7
> S;
[
    Permutation group acting on a set of cardinality 8
        (1, 8)(2, 3)(4, 5)(6, 7)
        (1, 3)(2, 8)(4, 6)(5, 7)
        (1, 5)(2, 6)(3, 7)(4, 8)
        (1, 2, 6, 3, 4, 5, 7),
    Permutation group acting on a set of cardinality 8
        (1, 8)(2, 3)(4, 5)(6, 7)
        (1, 3)(2, 8)(4, 6)(5, 7)
        (1, 5)(2, 6)(3, 7)(4, 8)
        (1, 2, 6, 3, 4, 5, 7)
        (1, 2, 3)(4, 6, 5),
    Permutation group acting on a set of cardinality 8
        (1, 2, 3, 4, 5, 6, 8)
        (1, 2, 4)(3, 6, 5)
        (1, 6)(2, 3)(4, 5)(7, 8),
    Permutation group acting on a set of cardinality 8
        (1, 2, 3, 4, 5, 6, 8)
        (1, 3, 2, 6, 4, 5)
        (1, 6)(2, 3)(4, 5)(7, 8),
    Permutation group acting on a set of cardinality 8
        (1, 8)(2, 3)(4, 5)(6, 7)
        (1, 3)(2, 8)(4, 6)(5, 7)
        (1, 5)(2, 6)(3, 7)(4, 8)
        (1, 2, 6, 3, 4, 5, 7)
        (1, 2, 3)(4, 6, 5)
        (1, 2)(5, 6),
    Permutation group acting on a set of cardinality 8
        (1, 2)(3, 4, 5, 6, 7, 8)
        (1, 2, 3),
    Permutation group acting on a set of cardinality 8
        (1, 2, 3, 4, 5, 6, 7, 8)
        (1, 2)
]

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Example: Non Conj (ex-974219)#

An example of isomorphic, non-conjugate transitive groups.

> G := TransitiveGroup(22, 26);
> H := TransitiveGroup(22, 27);
> b, _ := IsIsomorphic(G, H); b;
true
> S := SymmetricGroup(22);
> IsConjugate(S, G, H);
false

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Processes#

A transitive group process enables iteration over all transitive groups of specified degrees satisfying a given predicate, without having to create and store all such groups together. The intrinsic function TransitiveGroupProcess may be used to create a transitive group process in Magma. The standard process functions IsEmpty, Current, CurrentLabel and Advance can then be applied to the process.

TransitiveGroupProcess(d): RngIntElt -> Process#: Return a group process which will iterate though all transitive groups of degree \(d\).

TransitiveGroupProcess(S): [RngIntElt] -> Process#: Return a process which will iterate though all transitive groups with degree in the sequence \(S\).

TransitiveGroupProcess(d, f): RngIntElt, Program -> Process#: Return a process which will iterate though all transitive groups with degree \(d\) which satisfy the predicate \(f\).

TransitiveGroupProcess(S, f): [RngIntElt], Program -> Process#: Return a process which will iterate though all transitive groups with degree in the sequence \(S\) which satisfy the predicate \(f\).

IsEmpty(p): Process -> BoolElt#: Returns true if the process \(p\) has passed its last group.

Current(p): Process -> GrpPerm, MonStgElt#: Return the current group of the process \(p\), as well as a description of the group.

CurrentLabel(p): Process -> RngIntElt, RngIntElt#: Return the label of the current group of the process \(p\). That is, return \(d\) and \(n\) such that the current group is TransitiveGroup(d, n).

Advance(~p): Process#: Move the process \(p\) to its next group.

Example: Transitive Process (ex-851721)#

The use of processes is illustrated by the following code, in which the orders of all transitive groups of degree \(5\) are listed.

> p := TransitiveGroupProcess(5);
> while not IsEmpty(p) do
>     CurrentLabel(p), #Current(p);
>     Advance(~p);
> end while;
5 1 5
5 2 10
5 3 20
5 4 60
5 5 120

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Transitive Group Identification#

Given a transitive group \(G\) whose degree is at most 30, it is possible to obtain the number of the group in the transitive groups database which is isomorphic to \(G\).

TransitiveGroupIdentification(G): GrpPerm -> RngIntElt, RngIntElt#

Raw: BoolElt                    Default: true

The number (and degree) of the group in the transitive groups database which is isomorphic to the transitive group \(G\).

If the optional parameter Raw is set to \(\texttt{false}\), a third value is returned. In this case, the third value is a permutation conjugating the given group to the copy in the library.

Example: Transitive Id (ex-026a1e)#

We get a transitive permutation group from the small groups database and identify it as a transitive group.

> G := SmallGroup(336, IsTransitive: Search:="Insoluble");
> G;
Permutation group G acting on a set of cardinality 16
  (1, 14, 6, 2, 12, 8, 13, 7)(3, 15, 10, 5, 16, 9, 4, 11)
  (2, 5, 6)(3, 10, 9)(4, 15, 16)(7, 11, 13)
> TransitiveGroupIdentification(G : Raw := false);
715 16 (1, 16, 3, 4, 2, 11, 5, 6, 9, 8, 13, 10)(7, 12, 15)
> n, d, p := $1;
> G^p eq TransitiveGroup(d, n);
true

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We found it to be group 715 of degree 16.