# Atlas Data for the Sporadic Groups Most of the functions described here use data derived from the Web Atlas. The data has been prepared for inclusion in Magma by Michael Downward and Eamonn O’Brien. It maintains Atlas names, conventions and orderings. All of these functions, except `GoodBasePoints`, accept as input matrix or permutation groups. The algorithm underpinning `GoodBasePoints` due to O’Brien & Wilson [[O'Brien and Wilson, 2005](../../references.md#cite-obrien-wilson)]. ## `StandardGenerators(G, str : parameters): Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum` ```magma Projective : BoolElt Default: false AutomorphismGroup: BoolElt Default: false ``` Construct standard generators for small quasisimple or sporadic group $G$ having name *str*; words in SLP group defined on the defining generators of $G$ are also obtained for the standard generators. If $G$ is sporadic and `AutomorphismGroup` is `true`, assume $G$ is automorphism group of group having name *str*. If standard generators found, return `true` and sequences of generators and corresponding words, else `false`. **Note:** A return value of `false` only means that the algorithm’s random search for standard generators did not succeed within the number of tries allowed. If the user is sure the group $G$ matches the name *str*, then they should try the function again. If $G$ is absolutely irreducible matrix group and `Projective` is `true`, then construct standard generators possibly modulo centre of $G$. This function currently works for all sporadic simple groups and all quasisimple groups for which the simple quotient has order at most $2 \times 10^8$. If you call it with an invalid value of *str*, then it will print out a list of all valid values. ## `StandardGeneratorsGroupNames() -> SetIndx` A list of valid strings for the second argument of `StandardGenerators`. ## `StandardCopy(str): MonStgElt -> Grp, BoolElt` The standard copy of the group $G$ having the name `str`. If the second return value is `true`, then the group $H$ returned is a matrix group with nontrivial scalar subgroup $Z$, and it is $H/Z$ rather than $H$ that is isomorphic to $G$. ## `IsomorphismToStandardCopy(G, str : parameters): Grp, MonStgElt -> BoolElt, Map` ```magma Projective : BoolElt Default: false AutomorphismGroup: BoolElt Default: false ``` Use the `StandardGenerators` function to construct a (possibly projective) isomorphism from $G$ to a standard copy of $G$. Options as for `StandardGenerators`. The first returned value indicates whether the call of `StandardGenerators` was successful. ## `StandardPresentation(G, str : parameters): Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum` ```magma Projective : BoolElt Default: false Generators : SeqEnum Default: [] AutomorphismGroup: BoolElt Default: false ``` Return `true` if standard presentation is satisfied by generators of sporadic group $G$ having name *str*, else `false`. If `AutomorphismGroup` is `true`, assume $G$ is automorphism group of sporadic group having name *str*. Standard generators may be supplied as `Generators`, otherwise defining generators are assumed to be standard. If $G$ is absolutely irreducible matrix group and `Projective` is `true`, then verify presentation modulo centre of $G$. ## `MaximalSubgroups(G, str : parameters): Grp, MonStgElt -> BoolElt, SeqEnum, SeqEnum` ```magma Projective : BoolElt Default: false Generators : SeqEnum Default: [] AutomorphismGroup: BoolElt Default: false ``` Construct **some** maximal subgroups for sporadic group $G$ having name *str*. If `AutomorphismGroup` is `true`, assume $G$ is automorphism group of sporadic group having name *str* and construct **some** of its maximal subgroups. If standard generators supplied as `Generators` or found for $G$ then return `true` and list of subgroups, else return `false`. If $G$ is absolutely irreducible matrix group and `Projective` is `true`, then construct standard generators and so subgroups possibly modulo centre of G. ## `Subgroups(G, str : parameters): Grp, MonStgElt -> BoolElt, SeqEnum` ```magma Projective: BoolElt Default: false Generators: SeqEnum Default: [] ``` Construct certain subgroups for sporadic group $G$ having name *str*. If standard generators supplied as `Generators` or found for $G$ then return `true` and list of subgroups, else return `false`. If $G$ is absolutely irreducible matrix group and `Projective` is `true`, then construct standard generators possibly modulo centre of $G$. ## `GoodBasePoints(G, str : parameters): Grp, MonStgElt -> BoolElt, SeqEnum` ```magma Projective: BoolElt Default: false Generators: SeqEnum Default: [] ``` If standard generators supplied as `Generators` or found for sporadic group $G$ having name *str*, then return `true` and list of base points for $G$, else return `false`. If $G$ is absolutely irreducible and `Projective` is `true`, then standard generators are possibly modulo centre of $G$, and base points are correspondingly adjusted. ## `SubgroupsData(str): MonStgElt -> SeqEnum` Display stored subgroup data for sporadic group having name *str*. ## `MaximalSubgroupsData(str : parameters): MonStgElt -> SeqEnum` ```magma AutomorphismGroup: BoolElt Default: false ``` Display stored data for some maximal subgroups of sporadic group having name *str*. If `AutomorphismGroup` is `true`, then display stored data for some maximal subgroups of automorphism group of sporadic group. ## `Example: Sporadic J1 (ex-328794)` The machinery is illustrated in the case of the sporadic Janko group $J_1$. ```magma > G := > MatrixGroup<7, GF(11) | > [ 9, 1, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 9, 1, 3, 1, 3, 3, 9, 1, 3, 1, 3, > 3, 9, 1, 1, 1, 3, 3, 9, 1, 1, 3, 3, 3, 9, 1, 1, 3, 1, 3, 9, 1, 1, 3, 1, 3 ], > [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, > 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0,0] >; > flag, S := StandardGenerators (G, "J1"); > flag; true > StandardPresentation (G, "J1": Generators := S); true > flag, M:= MaximalSubgroups (G, "J1": Generators := S); > #M; 7 > M[4]; rec | name := 19:6, parent := J1, group := MatrixGroup(7, GF(11)) Generators: [ 0 1 4 3 3 4 7] [ 1 2 8 3 6 2 9] [ 4 8 10 1 6 0 9] [ 3 3 1 8 9 1 10] [ 3 6 6 9 1 3 7] [ 4 2 0 1 3 0 9] [ 7 9 9 10 7 9 0] [ 4 6 2 3 8 1 6] [ 8 1 3 10 2 7 4] [ 3 6 1 0 6 9 6] [ 2 3 6 9 0 3 7] [ 7 8 5 2 4 6 4] [10 4 5 2 8 6 8] [10 9 0 1 9 8 9], order := 114, index := 1540 > ```