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### 1 The SOTGrps package

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].

#### 1.1 Main functions

##### 1.1-1 AllSOTGroups
 ‣ 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 ] ) ]


##### 1.1-2 NumberOfSOTGroups
 ‣ 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.


##### 1.1-3 SOTGroup
 ‣ 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(n) without constructing all groups in the list. The option of constructing a PcpGroup is available for solvable groups.

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.

##### 1.1-4 IdSOTGroup
 ‣ 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(n,i). Note that if the input group is a PcpGroup, this may result in slow runtime, as IdSOTGroup may compute the Centre and/or the FittingSubgroup, which is slow for PcpGroups.

##### 1.1-5 IsIsomorphicSOTGroups
 ‣ 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


##### 1.1-6 IsSOTAvailable
 ‣ IsSOTAvailable( n ) ( function )

returns true if the order n is available in the SOTGrps library, and false otherwise.

##### 1.1-7 SOTGroupsInformation
 ‣ 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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