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#
gap> START_TEST("SOTGroupsInformation.tst");
#
gap> SOTGroupsInformation(5);
There is 1 group of order 5.
There is 1 cyclic group.
#
gap> SOTGroupsInformation(5^2);
There are 2 groups of order 25.
There is 1 cyclic group, and 1 elementary abelian group.
#
gap> SOTGroupsInformation(5^3);
There are 5 groups of order 125.
There are 3 abelian groups, and 2 extraspecial groups.
#
gap> SOTGroupsInformation(5^4);
There are 15 groups of order 625.
There are 5 abelian groups, and 10 nonabelian groups.
#
gap> SOTGroupsInformation(5*3);
There is 1 group of order 15.
There is 1 cyclic group.
#
gap> SOTGroupsInformation(5*11);
There are 2 groups of order 55.
There is 1 cyclic group, and 1 nonabelian group.
#
gap> SOTGroupsInformation(5^2*3);
There are 3 groups of order 75.
There are 2 abelian groups.
#
gap> SOTGroupsInformation(5^3*3);
There are 7 groups of order 375.
The groups of order p^3q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 3 are abelian.
SOT 4 - 5 are nonabelian nilpotent and have a normal Sylow 5-subgroup and
a normal Sylow 3-subgroup.
SOT 6 is non-nilpotent and has a normal Sylow 3-subgroup with Sylow
5-subgroup [ 125, 5 ].
SOT 7 is non-nilpotent and has a normal Sylow 3-subgroup with Sylow
5-subgroup [ 125, 3 ].
#
gap> SOTGroupsInformation(5^4*3);
There are 21 groups of order 1875.
The groups of order p^4q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 15 are nilpotent and all Sylow subgroups are normal.
SOT 16 is sovable, non-nilpotent and has a normal abelian Sylow 5-subgroup
[ 625, 2 ], with cyclic Sylow 3-subgroup.
SOT 17 is sovable, non-nilpotent and has a normal abelian Sylow 5-subgroup
[ 625, 11 ], with cyclic Sylow 3-subgroup.
SOT 18 - 19 are sovable, non-nilpotent and have a normal elementary
abelian Sylow 5-subgroup [ 625, 15 ], with cyclic Sylow
3-subgroup.
SOT 20 is sovable, non-nilpotent and has a normal nonabelian Sylow
5-subgroup [ 625, 14 ], with cyclic Sylow 3-subgroup.
SOT 21 is sovable, non-nilpotent and has a normal nonabelian Sylow
5-subgroup [ 625, 12 ], with cyclic Sylow 3-subgroup.
#
gap> SOTGroupsInformation(5^2*3^2);
There are 6 groups of order 225.
The groups of order p^2q^2 are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 4 are abelian and all Sylow subgroups are normal.
SOT 5 is non-abelian, non-nilpotent and has a normal Sylow 5-subgroup
[ 25, 2 ] with Sylow 3-subgroup [ 9, 1 ].
SOT 6 is non-abelian, non-nilpotent and has a normal Sylow 5-subgroup
[ 25, 2 ] with Sylow 3-subgroup [ 9, 2 ].
#
gap> SOTGroupsInformation(5^2*11^2);
There are 15 groups of order 3025.
The groups of order p^2q^2 are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 4 are abelian and all Sylow subgroups are normal.
SOT 5 is non-abelian, non-nilpotent and has a normal Sylow 11-subgroup
[ 121, 1 ] with Sylow 5-subgroup [ 25, 1 ].
SOT 6 is non-abelian, non-nilpotent and has a normal Sylow 11-subgroup
[ 121, 1 ] with Sylow 5-subgroup [ 25, 2 ].
SOT 7 - 10 are non-abelian, non-nilpotent and have a normal Sylow
11-subgroup [ 121, 2 ] with Sylow 5-subgroup [ 25, 1 ].
SOT 11 - 15 are non-abelian, non-nilpotent and have a normal Sylow
11-subgroup [ 121, 2 ] with Sylow 5-subgroup [ 25, 2 ].
#
gap> SOTGroupsInformation(5^2*3*11);
There are 5 groups of order 825.
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 275.
SOT 4 - 5 have Fitting subgroup of order 165.
#
gap> SOTGroupsInformation(13^3*3);
There are 19 groups of order 6591.
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 13-subgroup and a
normal Sylow 3-subgroup.
6 is non-nilpotent and has a normal Sylow 3-subgroup with Sylow
13-subgroup [ 2197, 1 ].
7 - 10 are non-nilpotent and have a normal Sylow 3-subgroup with Sylow
13-subgroup [ 2197, 2 ].
11 - 15 are non-nilpotent and have a normal Sylow 3-subgroup with Sylow
13-subgroup [ 2197, 5 ].
16 - 18 are non-nilpotent and have a normal Sylow 3-subgroup with Sylow
13-subgroup [ 2197, 3 ].
19 is non-nilpotent and has a normal Sylow 3-subgroup with Sylow
13-subgroup [ 2197, 4 ].
#
gap> SOTGroupsInformation(255025);
There are 32 groups of order 255025.
The groups of order p^2q^2 are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
1 - 4 are abelian and all Sylow subgroups are normal.
5 - 6 are non-abelian, non-nilpotent and have a normal Sylow 101-subgroup
[ 10201, 1 ] with Sylow 5-subgroup [ 25, 1 ].
7 is non-abelian, non-nilpotent and has a normal Sylow 101-subgroup
[ 10201, 1 ] with Sylow 5-subgroup [ 25, 2 ].
8 - 27 are non-abelian, non-nilpotent and have a normal Sylow 101-subgroup
[ 10201, 2 ] with Sylow 5-subgroup [ 25, 1 ].
28 - 32 are non-abelian, non-nilpotent and have a normal Sylow
101-subgroup [ 10201, 2 ] with Sylow 5-subgroup [ 25, 2 ].
#
gap> SOTGroupsInformation(2^3*3);
There are 15 groups of order 24.
The groups of order p^3q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 3 are abelian.
SOT 4 - 5 are nonabelian nilpotent and have a normal Sylow 2-subgroup and
a normal Sylow 3-subgroup.
SOT 6 is non-nilpotent and has a normal Sylow 2-subgroup [ 8, 1 ].
SOT 7 - 8 are non-nilpotent and have a normal Sylow 2-subgroup [ 8, 2 ].
SOT 9 is non-nilpotent and has a normal Sylow 2-subgroup [ 8, 5 ].
SOT 10 - 11 are non-nilpotent and have a normal Sylow 2-subgroup [ 8, 3 ].
SOT 12 is non-nilpotent and has a normal Sylow 2-subgroup [ 8, 4 ].
SOT 13 is non-nilpotent and has a normal Sylow 3-subgroup with Sylow
2-subgroup [ 8, 5 ].
SOT 15 is non-nilpotent, isomorphic to Sym(4), and has no normal Sylow
subgroups.
#
gap> SOTGroupsInformation(2^2*3^2);
There are 14 groups of order 36.
The groups of order p^2q^2 are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 4 are abelian and all Sylow subgroups are normal.
SOT 5 is non-abelian, non-nilpotent and has a normal Sylow 3-subgroup
[ 9, 1 ] with Sylow 2-subgroup [ 4, 1 ].
SOT 6 is non-abelian, non-nilpotent and has a normal Sylow 3-subgroup
[ 9, 1 ] with Sylow 2-subgroup [ 4, 2 ].
SOT 7 is non-abelian, non-nilpotent and has a normal Sylow 2-subgroup [4,
2] with Sylow 3-subgroup [9, 1].
SOT 8 - 10 are non-abelian, non-nilpotent and have a normal Sylow
3-subgroup [ 9, 2 ] with Sylow 2-subgroup [ 4, 1 ].
SOT 11 - 14 are non-abelian, non-nilpotent and have a normal Sylow
3-subgroup [ 9, 2 ] with Sylow 2-subgroup [ 4, 1 ].
#
gap> SOTGroupsInformation(2^4*3);
There are 52 groups of order 48.
The groups of order p^4q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 14 are nilpotent and all Sylow subgroups are normal.
SOT 15 is sovable, non-nilpotent and has a normal Sylow 3-subgroup, with
cylic Sylow 2-subgroup [ 16, 1 ].
SOT 16 - 17 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with abelian Sylow 2-subgroup [ 16, 5 ].
SOT 18 is sovable, non-nilpotent and has a normal Sylow 3-subgroup, with
abelian Sylow 2-subgroup [ 16, 2 ].
SOT 19 - 20 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with abelian Sylow 2-subgroup [ 16, 10 ].
SOT 21 is sovable, non-nilpotent and has a normal Sylow 3-subgroup, with
elementary abelian Sylow 2-subgroup [ 16, 14 ].
SOT 22 - 24 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 13 ].
SOT 25 - 27 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 11 ].
SOT 28 - 29 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 3 ].
SOT 30 - 31 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 12 ].
SOT 32 - 33 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 4 ].
SOT 34 - 35 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 6 ].
SOT 36 - 38 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 8 ].
SOT 39 - 40 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 7 ].
SOT 41 - 42 are sovable, non-nilpotent and have a normal Sylow 3-subgroup,
with nonabelian Sylow 2-subgroup [ 16, 9 ].
SOT 43 is sovable, non-nilpotent and has a normal abelian Sylow 2-subgroup
[ 16, 2 ], with cyclic Sylow 3-subgroup.
SOT 44 is sovable, non-nilpotent and has a normal abelian Sylow 2-subgroup
[ 16, 10 ], with cyclic Sylow 3-subgroup.
SOT 45 - 46 are sovable, non-nilpotent and have a normal elementary
abelian Sylow 2-subgroup [ 16, 14 ], with cyclic Sylow 3-subgroup.
SOT 47 is sovable, non-nilpotent and has a normal nonabelian Sylow
2-subgroup [ 16, 13 ], with cyclic Sylow 3-subgroup.
SOT 48 is sovable, non-nilpotent and has a normal nonabelian Sylow
2-subgroup [ 16, 12 ], with cyclic Sylow 3-subgroup.
SOT 49 - 52 are solvable, non-nilpotent, and have no normal Sylow
subgroups.
#
gap> SOTGroupsInformation(3^4*13);
There are 51 groups of order 1053.
The groups of order p^4q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 15 are nilpotent and all Sylow subgroups are normal.
SOT 16 is sovable, non-nilpotent and has a normal Sylow 13-subgroup, with
cylic Sylow 3-subgroup [ 81, 1 ].
SOT 17 - 18 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with abelian Sylow 3-subgroup [ 81, 5 ].
SOT 19 is sovable, non-nilpotent and has a normal Sylow 13-subgroup, with
abelian Sylow 3-subgroup [ 81, 2 ].
SOT 20 - 21 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with abelian Sylow 3-subgroup [ 81, 11 ].
SOT 22 is sovable, non-nilpotent and has a normal Sylow 13-subgroup, with
elementary abelian Sylow 3-subgroup [ 81, 15 ].
SOT 23 - 25 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 14 ].
SOT 26 - 28 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 6 ].
SOT 29 - 32 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 13 ].
SOT 33 - 34 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 3 ].
SOT 35 - 37 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 4 ].
SOT 38 - 39 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 12 ].
SOT 40 - 42 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 8 ].
SOT 43 - 44 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 9 ].
SOT 45 - 47 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 7 ].
SOT 48 - 49 are sovable, non-nilpotent and have a normal Sylow
13-subgroup, with nonabelian Sylow 3-subgroup [ 81, 10 ].
SOT 50 is sovable, non-nilpotent and has a normal elementary abelian Sylow
3-subgroup [ 81, 15 ], with cyclic Sylow 13-subgroup.
SOT 51 is solvable, non-nilpotent, and has no normal Sylow subgroups.
#
gap> SOTGroupsInformation(2^2*3*5);
There are 13 groups of order 60.
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 - 5 have Fitting subgroup of order 15.
SOT 6 has Fitting subgroup of order 20.
SOT 7 - 12 have Fitting subgroup of order 30.
SOT 13 is nonsolvable and has Fitting subgroup of order 1.
#
gap> SOTGroupsInformation(3^4*5);
There are 16 groups of order 405.
The groups of order p^4q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 15 are nilpotent and all Sylow subgroups are normal.
SOT 16 is sovable, non-nilpotent and has a normal elementary abelian Sylow
3-subgroup [ 81, 15 ], with cyclic Sylow 5-subgroup.
#
gap> SOTGroupsInformation(2^4*17);
There are 54 groups of order 272.
The groups of order p^4q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
SOT 1 - 14 are nilpotent and all Sylow subgroups are normal.
SOT 15 - 18 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with cylic Sylow 2-subgroup [ 16, 1 ].
SOT 19 - 23 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with abelian Sylow 2-subgroup [ 16, 5 ].
SOT 24 - 25 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with abelian Sylow 2-subgroup [ 16, 2 ].
SOT 26 - 28 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with abelian Sylow 2-subgroup [ 16, 10 ].
SOT 29 is sovable, non-nilpotent and has a normal Sylow 17-subgroup, with
elementary abelian Sylow 2-subgroup [ 16, 14 ].
SOT 30 - 32 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 13 ].
SOT 33 - 35 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 11 ].
SOT 36 - 38 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 3 ].
SOT 39 - 40 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 12 ].
SOT 41 - 43 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 4 ].
SOT 44 - 47 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 6 ].
SOT 48 - 50 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 8 ].
SOT 51 - 52 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 7 ].
SOT 53 - 54 are sovable, non-nilpotent and have a normal Sylow
17-subgroup, with nonabelian Sylow 2-subgroup [ 16, 9 ].
#
gap> SOTGroupsInformation(19*23*29*31);
There is 1 group of order 392863.
All groups of order 392863 are abelian.
#
gap> SOTGroupsInformation(11*23*29*31);
There are 2 groups of order 227447.
The groups of order pqrs are solvable and classified by O. H"older.
These groups are sorted by their centre.
SOT 1 is abelian.
SOT 2 has centre of order that is a product of two distinct primes.
#
gap> SOTGroupsInformation(3*7*43*3613);
There are 61 groups of order 3262539.
The groups of order pqrs are solvable and classified by O. H"older.
These groups are sorted by their centre.
SOT 1 is abelian.
SOT 2 - 7 have centre of order that is a product of two distinct primes.
SOT 8 - 23 have a cyclic centre of prime order.
SOT 24 - 61 have a trivial centre.
#
gap> STOP_TEST("SOTGroupsInformation.tst", 1);
[ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
]
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