Database of Simple Groups

The database of simple groups of order less than 10²⁰ is intended to make it easy for a user to access any of the small simple groups. There are 403,874 simple groups in the database. The groups are arranged in order of increasing cardinality which makes it trivial to scan the simple groups of small order.

SimpleGroup(N) : RngIntElt -> Grp

SimpleGroup(S) : MonStgElt -> Grp

SimpleGroup(T) : Tup -> Grp

This intrinsic returns a group isomorphic to the simple group specified in the argument. The simple group can be specified by any one of the following:-

-

An integer N that is the number of the requested simple group in the database.

-

A string S giving the standard name of the simple group.

-

A tuple T containing three integers that give the internal designation of the simple group.

The group returned is generally a permutation group of smallest degree that is isomorphic to the simple group, but there are cases where a matrix group is returned.

SimpleGroupOfOrder(M) : RngIntElt -> Grp

SimpleGroup(F) : RngIntEltFac -> Grp

A group isomorphic to a simple group whose order is given by the argument. The order may be an integer M or a factored integer F.

NumberOfSimpleGroups() : -> RngIntElt

The number of groups in the simple groups database.

SimpleGroupName(N) : RngIntElt -> MonStgElt

This intrinsic returns a standard name of the simple group which has database number N.

SimpleGroupNameToNumber(S) : MonStgElt -> RngIntElt

This intrinsic returns the database number of the simple group which has standard name S.

SimpleGroupIDToNumber(T) : Tup -> RngIntElt

This intrinsic returns the database number of the simple group whose internal designation is T.

SimpleGroupID(N) : RngIntElt -> Tup

This intrinsic returns the tuple identifying the simple group with database number N.