Magma includes representations of nearly simple groups from the ATLAS of Finite Group Representations http://web.mat.bham.ac.uk/atlas/v2.0. The data was supplied by Robert Wilson.
Groups in the database are accessed by name. The intrinsic ATLASGroupNames gives a list of the names that may currently be used to access the database. The names are based on ATLAS names for simple groups, with some exceptions (usually caused by an aversion to subscripting automorphisms). Classical group names take precedence over their Lie-type names. Within a name, the letter "T" denotes a twisted group of Lie type. (The two sorts of twisting of D4 are distinguished by one being "O8m" and the other "TD4".) An initial number on the name denotes a central element, a "d" is used to separate the simple group name from an automorphism (when there is no other letter there), and an "i" denotes an isoclinic variant.
The basic access function takes a name and returns a special type of group, an ATLAS group, with Magma type GrpAtlas. Access to the information stored about the named group are then done through this ATLAS group.
> ATLASGroupNames();
{@ A5, 2A5, 2S5, 2S5i, S5, A6, 2A6, 2S6, 3A6, 3S6, 6A6,
6S6, A6V4, M10, PGL29, S6, A7, A8, 2A8, S8, A9, 2A9,
S9, A10, 2A10, S10, A11, 2A11, 2S11, S11, A12, 2A12,
S12, A13, 2A13, S13, A14, 2A14, 2S14, 2S14i, S14, O93,
2O93, 2O93d2, O93d2, O10m2, O10m2d2, O73, 2O73, 2O73d2,
3O73, 3O73d2, O73d2, O8m2, O8m2d2, O8m3, 2O8m3,
2O8m3d2a, O8m3D8, O8m3V4, O8m3d2a, O8p2, S102, S44,
S44d2, S44d4, S45, 2S45, S45d2, S47, 2S47, 2S47d2,
S47d2, S62, 2S62, S63, 2S63, 2S63d2, S63d2, S82, U311,
3U311, 3U311d2, U311d2, U33, U33d2, U42, 2U42, 2U42d2,
U42d2, U43, U52, U52d2, U53, U62, 12U62, 2U62, 3U62,
4U62, 6U62, U62S3, U62d2, U72, E74, E85, E82, E72, E62,
TF42, TF42d2, G25, TE62, 2TE62, 2TE62d2, 3TE62,
3TE62S3, 3TE62d2, 3TE62d3, 4TE62, TE62S3, TE62d2,
TE62d3, E64, 3E64, 3E64d2, TD42, TD42d3, G23, 3G23,
3G23d2, G23d2, G24, 2G24, 2G24d2, 2G24d2i, G24d2, F42,
2F42, 2F42d2, 2F42d4i, F42d2, R27, R27d3, Sz8, 2Sz8,
4Sz8d3, Sz8d3, Sz32, Sz32d5, TD43, L27, L28, L28d3,
L211, 2L211, L211d2, L213, 2L213, 2L213d2, L213d2,
L216, L216d2, L216d4, L217, 2L217, 2L217d2, L217d2,
L219, 2L219, 2L219d2i, L219d2, L223, 2L223, 2L223d2i,
L223d2, L227, L229, 2L229, L231, 2L231, L231d2, L232,
L232d5, L249, 2L249, L33, L33d2, L34, 12aL34, 12bL34,
2L34, 3L34, 4aL34, 4bL34, 6L34, L35, L35d2, L37, 3L37,
3L37d2, L37d2, L311, L52, L52d2, L62, L62d2, L72,
L72d2, B, Co1, 2Co1, Co2, Co3, F22, 2F22, 2F22d2, 3F22,
3F22d2, F22d2, F23, F24, 3F24, 3F24d2, F24d2, HN, HNd2,
HS, 2HS, 2HSd2, HSd2, He, Hed2, J1, J2, 2J2, 2J2d2,
J2d2, J3, 3J3, 3J3d2, J3d2, J4, Ly, ON, 3ON, 3ONd2,
ONd2, ONd4, Ru, 2Ru, Suz, 2Suz, 2Suzd2, 3Suz, 3Suzd2,
6Suz, 6Suzd2, Suzd2, Th, M, M11, M12, 2M12, 2M12d2,
M12d2, M22, 12M22, 2M22, 2M22d2, 3M22, 3M22d2, 4M22,
4M22d2, 6M22, 6M22d2, M22d2, M23, M24, McL, 3McL,
3McLd2, McLd2, S7 @}
The names of the groups that have representations stored in the database.
The ATLAS group stored in the database that has name N.
Once an ATLAS group has been extracted from the database, the following intrinsics give access to the information stored with it.
The order of A.
The order of the multiplier of A, when A is simple.
The sequence of keys to the matrix representations of A stored in the database. This will be the empty sequence if no matrix representations are stored.
The set of degrees of the matrix representations stored for A.
The degree of the matrix representation associated with key K.
The set of sizes of the fields for which a matrix representation of A is available.
The set of characteristics of the fields for which a matrix representation of A is available.
The base field of the matrix representation associated with key K.
The sequence of keys to the permutation representations of A stored in the database. This will be the empty sequence if no permutation representations are stored.
The set of degrees of the permutation representations stored for A.
The degree of the permutation representation associated with key K.
The intrinsics described below construct concrete representations of the ATLAS groups from the data in the database. Each representation is accessed by its key, sequences of which are produced by the intrinsics MatRepKeys and PermRepKeys described above. The intrinsics described in this section take a key and produce a concrete representation.
Given a key to a matrix representation of an ATLAS group, construct and return the corresponding matrix group.
The generators of the matrix group designated by database key K.
Given a key to a permutation representation of an ATLAS group, construct and return the corresponding permutation group.
The generators of the permutation group designated by database key K.
> A := ATLASGroup("2J2d2");
> PermRepKeys(A);
[]
> mrk := MatRepKeys(A);
> mrk;
[
Matrix rep of degree 12 over GF(3),
Matrix rep of degree 6 over GF(25) named a,
Matrix rep of degree 12 over GF(7)
]
The database has no permutation representations and three matrix
representations. We construct the first of the matrix groups. It is
small enough to check its composition factors.
> K := mrk[1];
> M := MatrixGroup(K);
> M`Order := #A;
> RandomSchreier(M);
> CompositionFactors(M);
G
| Cyclic(2)
*
| J2
*
| Cyclic(2)
1
For efficiency, we asserted the order of the matrix group to be the
order of the ATLAS group and constructed a BSGS by the random schreier.