This chapter describes procedures to access the information stored in the Group Presentations Library. Currently, all presentations stored are presentations of Sporadic Groups. We consider a sporadic group G to be a sporadic simple group or an automorphism group of a sporadic simple group. At the moment, the library contains 19 presentations of simple sporadic groups and 8 presentations of non-simple sporadic groups. Some of these presentations present isomorphic groups. All of the presentations currently available can be found in [PS97]. Especially, presentations are present for the seven smallest sporadic simple groups and their automorphism groups. For every maximal subgroup in these eleven groups, short words are given that generate the maximal subgroup in the given presentation. Maximal is considered here as up to the action of the automorphism group of the group presented.
Information about some particular sporadic group can be obtained by using its name. The names that appear in the following example are the names that are used to identify the sporadic groups. These names correspond with the names used in GAP to identify character tables.
gap> Read("gpl.g");
gap> av := AvailableGroupPresentationsGPL();
[ GroupPresentationGPL( "M11", 1 ), GroupPresentationGPL( "M12", 1 ),
GroupPresentationGPL( "M12:2", 1 ), GroupPresentationGPL( "J1", 1 ),
GroupPresentationGPL( "M22", 1 ), GroupPresentationGPL( "M22:2", 1 ),
GroupPresentationGPL( "J2", 1 ), GroupPresentationGPL( "J2:2", 1 ),
GroupPresentationGPL( "M23", 1 ), GroupPresentationGPL( "HS", 1 ),
GroupPresentationGPL( "HS:2", 1 ), GroupPresentationGPL( "M24", 1 ),
GroupPresentationGPL( "M24", 2 ), GroupPresentationGPL( "McL", 1 ),
GroupPresentationGPL( "He", 1 ), GroupPresentationGPL( "He:2", 1 ),
GroupPresentationGPL( "Suz", 1 ), GroupPresentationGPL( "Suz:2", 1 ),
GroupPresentationGPL( "ON", 1 ), GroupPresentationGPL( "Co3", 1 ),
GroupPresentationGPL( "Co2", 1 ), GroupPresentationGPL( "Fi22", 1 ),
GroupPresentationGPL( "Fi22:2", 1 ),
GroupPresentationGPL( "Fi23", 1 ), GroupPresentationGPL( "Co1", 1 ),
GroupPresentationGPL( "Co1", 2 ),
GroupPresentationGPL( "F3+:2", 1 ) ]
gap> for i in [1..27] do InitializeGroupPresentationGPL( av[i] ); od;
gap> List(av, p -> AvailableSubgroupsGPL( p ));
[ [ A6.2_3, L2(11), A6, 3^2:Q8.2, A5.2, 3^2:8, 2.S4 ],
[ M11, A6.2^2, PGL2(9), S6, L2(11), L2(11), 3^2.2.S4, 2xS5, M8.S4,
4^2:D12, A4xS3 ],
[ L2(11):2, L2(11):2, (2^2xA5):2, 3_{+}^{1+2}:D8, (M8.S4).2,
(4^2:D12).2, S4xS3, S5 ],
[ L2(11), 2^3.7.3, 2xA5, 19:6, 11:10, D6xD10, 7:6 ],
[ L3(4), 2^4:a6, A7, 2^4:s5, 2^4:L2(5), 2^3:sl(3,2), A6.2_3,
L2(11) ],
[ L3(4):2_2, 2^4:s6, 2^5:s5, 2^5:L2(5), 2^3:sl(3,2)x2, A6.2^2,
L2(11):2 ],
[ U3(3), 3.A6.2_2, 2^1+4b:a5, 2^2+4.3xs3, a4xa5, a5xd10, L3(2).2,
5^2:d12, A5 ],
[ U_3(3):2, 3.A6.2^2, 2^1+4b.S5, 2^2+4:3xS3.2, (A4xA5):2, (A5xD10).2,
L3(2):2x2, 5^2:(4xS3), S5 ],
[ M22, L3(4).2_2, 2^4:a7, A8, M11, 2^4:(3xa5):2, 23:11 ],
[ M22, U3(5).2, U3(5), L3(4).2_1, A8.2, 2^4.s6, 4^3:psl(3,2), M11,
4.2^4:s5, 2xa6.2^2, 5:4xa5 ],
[ M22:2, L3(4).2_2, A8.2x2, 2^5.s6, 4^3:(psl(3,2)x2), 2_+^(1+6):s5,
(2xa6.2^2).2, 5_+^{1+2}:[2^5], 5:4xs5 ],
[ M23, M22:2, 2^4:A8, L3(4):S3 ],
[ M12:2, 2^6:3:S6, M12, 2^6:3:A6, 2^6:(L3(2)xS3) ],
[ U4(3), M22, U3(5), 3^{1+4}_+:2.S5 ], [ S4(4):2 ], [ S4(4):4 ],
[ G2(4), 3_2.U4(3):2_3 ], [ 3.U4(3).(2^2) ], [ L3(7):2 ],
[ McL:2, McL, HS, M23, 2.S6(2) ],
[ U6(2):2, U6(2), 2^{10}:M22:2, McL, 2^{1+8}:S6(2), HS:2 ],
[ 2.U6(2), O7(3), 2^{10}:M22, 2^6:S6(2) ], [ O8^+(2):S3x2 ],
[ 2.Fi22, O8^+(3):S3, O8^+(3):3, O8^+(3):2 ], [ Co2, 2^{11}:M24,
Co3 ],
[ 2x3.Suz:2 ], [ Fi23x2 ] ]
Note that to be able to work with a presentation it has to be initialized first using the procedure InitializeGroupPresentationGPL The subgroups are ordered according to their size. In case of equal size, maximal subgroups are listed before non-maximal ones.
The first section in this chapter describes the function GroupRecordGPL that returns a record containing all available presentations of a group. The next sections describe functions that initialize and access the presentations themselves. The final sections describe functions that extract certain subgroups in a given presentation from the library.