With some overlaps, the **SOTGrps** package extends the Small Group Library to give access to some more "small" orders. For example, it constructs a complete and irredundant list of isomorphism type representatives of the groups of order

that factorises into at most four primes;

\(p^4q\), for distinct primes \(p\) and \(q\).

The mathematical background for this package is described in [DEP22].

`‣ AllSOTGroups` ( n[, filter] ) | ( function ) |

takes in a number `n` that factorises into at most four primes or is of the form \(p^4q\) (\(p\), \(q\) are distinct primes), and returns a complete and duplicate-free list of isomorphism class representatives of the groups of order `n`. Solvable groups are using refined polycyclic presentations. By default, solvable groups are constructed in the filter `IsPcGroup`

, but if the optional argument `filter` is set to `IsPcpGroup`

then the groups are constructed in that filter instead. Nonsolvable groups are always returned as permutation groups.

gap> AllSOTGroups(60); [ <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>, Alt( [ 1 .. 5 ] ) ]

`‣ NumberOfSOTGroups` ( n ) | ( function ) |

takes in a number `n` that factorises into at most four primes or of the form \(p^4q\) (\(p\), \(q\) are distinct primes), and returns the number of isomorphism types of groups of order `n`.

gap> NumberOfSOTGroups(2*3*5*7); 12 gap> NumberOfSOTGroups(2*3*5*7*11); Error, Order 2310 is not supported by SOTGrps. Please refer to the SOTGrps documentation for the list of supported orders.

`‣ SOTGroup` ( n, i[, arg] ) | ( function ) |

takes in a pair of numbers `n, i`, where `n` factorises into at most four primes or of the form \(p^4q\) (\(p\), \(q\) are distinct primes), and returns the `i`-th group with respect to the ordering of the list `AllSOTGroups(`

without constructing all groups in the list. The option of constructing a PcpGroup is available for solvable groups.`n`)

gap> SOTGroup(2*3*5*7, 1); <pc group of size 210 with 4 generators>

If the input `i` exceeds the number of groups of order `n`, an error message is returned.

`‣ IdSOTGroup` ( G ) | ( attribute ) |

takes in a group of order determines the SOT library number of `G`; that is, the function returns a pair [`n`, `i`] where `G` is isomorphic to `SOTGroup(`

. Note that if the input group is a PcpGroup, this may result in slow runtime, as `n`,`i`)`IdSOTGroup`

may compute the `Centre`

and/or the `FittingSubgroup`

, which is slow for PcpGroups.

`‣ IsIsomorphicSOTGroups` ( G, H ) | ( function ) |

determines whether two groups `G`, `H` are isomorphic. It is assumed that the input groups are available in the **SOTGrps** library.

gap> G:=Image(IsomorphismPermGroup(SmallGroup(690,1)));; gap> H:=Image(IsomorphismPcGroup(SmallGroup(690,1)));; gap> IsIsomorphicSOTGroups(G,H); true

`‣ IsSOTAvailable` ( n ) | ( function ) |

returns `true`

if the order `n` is available in the **SOTGrps** library, and `false`

otherwise.

`‣ SOTGroupsInformation` ( n ) | ( function ) |

prints information on the groups of the specified order. Since there are some overlaps between the existing SmallGrps library and the **SOTGrps** library. In particular, **SOTGrps** may construct the groups in a different order and so generate a different group ID; we denote such IDs by `SOT`

. If the order covered in **SOTGrps** library has no conflicts with the existing library, then such a flag is removed.

gap> SOTGroupsInformation(2^2*3*19); There are 15 groups of order 228. The groups of order p^2qr are either solvable or isomorphic to Alt(5). The solvable groups are sorted by their Fitting subgroup. SOT 1 - 2 are the nilpotent groups. SOT 3 has Fitting subgroup of order 57. SOT 4 - 7 have Fitting subgroup of order 76. SOT 8 - 9 have Fitting subgroup of order 38. SOT 10 - 15 have Fitting subgroup of order 114. gap> SOTGroupsInformation(2662); There are 15 groups of order 2662. The groups of order p^3q are solvable by Burnside's pq-Theorem. These groups are sorted by their Sylow subgroups. 1 - 3 are abelian. 4 - 5 are nonabelian nilpotent and have a normal Sylow 11-subgroup and a normal Sylow 2-subgroup. 6 is non-nilpotent and has a normal Sylow 2-subgroup [ 2, 1 ] with Sylow 11-subgroup [ 1331, 1 ]. 7 - 9 are non-nilpotent and have a normal Sylow 2-subgroup [ 2, 1 ] with Sylow 11-subgroup [ 1331, 2 ]. 10 - 12 are non-nilpotent and have a normal Sylow 2-subgroup [ 2, 1 ] with Sylow 11-subgroup [ 1331, 5 ]. 13 - 14 are non-nilpotent and have a normal Sylow 2-subgroup [ 2, 1 ] with Sylow 11-subgroup [ 1331, 3 ]. 15 is non-nilpotent and has a normal Sylow 2-subgroup [ 2, 1 ] with Sylow 11-subgroup [ 1331, 4 ].

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