# Introduction

This chapter describes the use of the various databases of groups that form part of Magma. The available databases are as follows:

*Simple Groups*: This database contains all the simple groups of order less than $10^{20}$. While Magma contains tools that allow the user to construct any finite simple group the purpose of this facility is to make it easier to access a simple group and to run through the simple groups in order of increasing group order.

*Small Groups*: This database is constructed by Hans Ulrich Besche, Heiko Dietrich, Bettina Eick, Eamonn O’Brien, and Eileen Pan [[Besche and Eick, 1999](../../references.md#cite-besche-eick-constr), [Besche and Eick, 1999](../../references.md#cite-besche-eick-1000), [Besche and Eick, 2001](../../references.md#cite-besche-eick-order-qnp), [Besche *et al.*, 2001](../../references.md#cite-besche-eick-obrien-order-2000), [Dietrich and Eick, 2005](../../references.md#cite-dietrich-eick), [Dietrich *et al.*, 2022](../../references.md#cite-dep4primes), [M.F. Newman and Vaughan-Lee, 2004](../../references.md#cite-newman-ob-vl-04), [O'Brien, 1990](../../references.md#cite-obrien-pgroup), [O'Brien, 1991](../../references.md#cite-obrien-groups-256), [O'Brien and Vaughan-Lee, 2005](../../references.md#cite-ob-vl-05)], and contains the following groups:

- All groups of order up to 2000, excluding the groups of order 1024.

- The groups whose order is the product of at most 4 primes.

- The groups of order dividing $p^7$ for $p$ a prime.

- The groups of order $3^8$.

- The groups of order $q^n p$, where $q^n$ is a prime-power dividing $2^8$, $3^6$, $5^5$ or $7^4$ and $p$ is a prime different to $q$.

- The groups of square-free order.

For a different mechanism for accessing the $p$-groups in this collection, see Section [The $p$-groups of Order Dividing $p^7$](p-groups.md#chapgrppgp), specifically the functions `SearchPGroups` and `CountPGroups`. These functions also access groups of order $p^7$.

$p$*-groups*: Magma contains the means to construct all $p$-groups of order $p^n$ where $n\le 7$. The data used in the constructions was supplied by Hans Ulrich Besche, Bettina Eick, Eamonn O’Brien, Mike Newman and Michael Vaughan-Lee [[Besche and Eick, 1999](../../references.md#cite-besche-eick-constr), [Besche and Eick, 1999](../../references.md#cite-besche-eick-1000), [Besche and Eick, 2001](../../references.md#cite-besche-eick-order-qnp), [Besche *et al.*, 2001](../../references.md#cite-besche-eick-obrien-order-2000), [M.F. Newman and Vaughan-Lee, 2004](../../references.md#cite-newman-ob-vl-04), [O'Brien, 1990](../../references.md#cite-obrien-pgroup), [O'Brien, 1991](../../references.md#cite-obrien-groups-256), [O'Brien and Vaughan-Lee, 2005](../../references.md#cite-ob-vl-05)].

*Metacyclic* $p$*-groups*: Magma is able to construct all metacyclic groups of order $p^n$. This machinery was developed by Mike Newman, Eamonn O’Brien, and Michael Vaughan-Lee.

*Perfect Groups*: This database contains all perfect groups up to order 50000, and many classes of perfect groups up to order one million. Each group is defined by means of a finite presentation. Further information is also provided which allows the construction of permutation representations. This database was constructed by Derek Holt and Willem Plesken [[Holt and Plesken, 1989](../../references.md#cite-holt-plesken)].

*Almost Simple Groups:* This database contains information about every group $G$, where $S \leq G \leq {\operatorname{Aut}}(S)$ and $S$ is a simple group of order less than 16000000, or $S$ is one of $M_{24}$, $HS$, $J_3$, $McL$, $Sz(32)$ or $L_6(2)$.

*Transitive Permutation Groups*: This database is a Magma version of the database of transitive permutation groups constructed by A. Hulpke [[Hulpke, 2005](../../references.md#cite-hulpketrans)] (for degree up to 30), J. Cannon and D. Holt [[Cannon and Holt, 2008](../../references.md#cite-trans32)] (degree 32), D. Holt and G. Royle [[Holt and Royle, 2019](../../references.md#cite-holtroyle)] (degrees 33 to 47), and D. Holt (degree 48). It contains all transitive permutation groups, up to conjugacy in the ambient symmetric group, having degree up to 48.

*Primitive Permutation Groups*: This is a database containing all primitive permutation groups having degree up to 8191 as determined by Sims (for degree $\le 50$), Roney-Dougal and Unger [[Roney-Dougal and Unger, 2003](../../references.md#cite-prim1000)] (for degree $< 1000$), Roney-Dougal [[Roney-Dougal, 2005](../../references.md#cite-prim2500)] (for degree $< 2500$), Coutts, Quick, and Roney-Dougal [[Coutts *et al.*, 2011](../../references.md#cite-prim4096)] (for degree $< 4096$), and Stratford (for degree $< 8192$).

*Rational Maximal Matrix Groups*: This contains the rational maximal finite matrix groups and their invariant forms, for small dimensions (up to 31) as determined by Gabi Nebe and Willem Plesken [[Nebe, 1996](../../references.md#cite-nebe-ratgps-31), [Nebe and Plesken, 1995](../../references.md#cite-nebe-plesken-ratgps)]. Each entry can be accessed either as a matrix group or as a lattice.

*Quaternionic Matrix Groups*: A database of the finite absolutely irreducible subgroups of ${\operatorname{GL}}_n({\cal D})$ where ${\cal D}$ is a definite quaternion algebra whose centre has degree $d$ over ${\mathbb{Q}}$ and $nd\leq10$. Each entry can be accessed either as a matrix group or as a lattice. The database was constructed by Gabi Nebe [[Nebe, 1998](../../references.md#cite-nebe-quaternionic)].

*Irreducible Matrix Groups*: A database of the irreducible subgroups of ${\operatorname{GL}}_n(p)$, $p$ prime, $n \ge 1$ and $p^n < 2500$. The groups were determined by Colva Roney-Dougal and William Unger [[Roney-Dougal and Unger, 2003](../../references.md#cite-prim1000)] (for $p^n < 1000$) and Roney-Dougal [[Roney-Dougal, 2005](../../references.md#cite-prim2500)].

*Soluble Irreducible Groups*: This database contains one representative of each conjugacy class of irreducible soluble subgroups of ${\rm GL}(n,p)$, $p$ prime, for $n > 1$ and $p^n < 256$. It was constructed by Mark Short [[Short, 1992](../../references.md#cite-short)].

*ATLAS Groups*: This database contains representations of nearly simple groups, as in the Birmingham ATLAS of Finite Group Representations. The data was supplied by Rob Wilson.

*Fundamental Groups of 3-Manifolds*: This database consists of the fundamental groups of the 10,986 small-volume closed hyperbolic manifolds in the Hodgson–Weeks census.

*Automatic Groups of 3-Manifolds*: This database contains automatic groups for 5,389 of the 10,986 small-volume closed hyperbolic manifolds in the Hodgson–Weeks census.