# 58 Automorphism Groups of Special Ag Groups

This chapter describes functions which compute and display information about aut``` groups of finite soluble groups. ```

``` The algorithm used for computing the aut``` group requires that the soluble group be given in terms of a special ag presentation. Such presentations are described in the chapter of the GAP manual which deals with ```Special Ag Groups```. Given a group presented by an arbitrary ag presentation, a special ag presentation can be computed using the function `SpecialAgGroup`.

The aut` group is returned as a standard GAP group record. Aut`s are represented by their action on the sag group generating set of the input group. The order of the aut``` group is also computed. ```

``` The performance of the aut``` group algorithm is highly dependent on the structure of the input group. Given two groups with the same sequence of LG-series factor groups it will usually take much less time to compute the aut` group of the one with the larger aut` group. For example, it takes less than 1 second (Sparc 10/52) to compute the aut``` group of the exponent 7 extraspecial group of order 7^3. It takes more than 40 seconds to compute the aut``` group of the exponent 49 extraspecial group of order 7^3. The orders of the aut``` groups are 98784 and 2058 respectively. It takes only 20 minutes (Sparc 10/52) to compute the aut``` group of the 2-generator Burnside group of exponent 6, a group of order 2^{28}cdot 3^{25} whose aut``` group has order 2^{40}cdot 3^{53}cdot 5cdot 7; note, however, that it can take substantially longer than this to compute the aut``` groups of some of the groups of order 64 (for nilpotent groups one should use the function `AutomorphismsPGroup` from the ANU PQ package instead).

The following section describes the function that computes the aut``` group of a sag``` (see AutGroupSagGroup). It is followed by a description of Automorphism Group Elements and Operations for Automorphism Group Elements). Functions for obtaining some structural information about the aut group are described next (see AutGroupStructure, AutGroupFactors and AutGroupSeries). Finally, a function that converts the aut``` group into a form which may be more suitable for some applications is described (see AutGroupConverted). ```

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### ` Subsections`

``` AutGroupSagGroup Automorphism Group Elements Operations for Automorphism Group Elements AutGroupStructure AutGroupFactors AutGroupSeries AutGroupConverted Previous Up NextIndex GAP 3.4.4April 1997```