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## Construction of p-groups of order dividing p^4
## Such low-power p-groups are well-known; our construction uses iterative extensions.
## For further details of the classification of groups of order p^4, we also refer to Alder, Garlow, Wheland's paper "Groups of order p^4 made less difficult" arXiv:1611.00461
####################################
SOTRec.lowpowerPGroups := function(p, k, arg...)
local PG, P2, P3, order8, r, P4, order16, order81, list;
if p > 1 and k = 1 then
if Length(arg) = 0 then
list := [SOTRec.groupFromData([ [p] ])];
else
list := [ [ [p] ] ];
fi;
return list;
fi;
if k = 2 then
P2 := [ [ [p, p], [1, [2, 1]] ], [ [p, p] ] ];
if Length(arg) = 0 then
list := List(P2, x -> SOTRec.groupFromData(x));
else
list := P2;
fi;
return list;
fi;
if p > 2 and k = 3 then
P3 := [ [ [p, p, p], [1, [2, 1]], [2, [3, 1]] ], [ [p, p, p], [1, [2, 1]] ], [ [p, p, p] ], [ [p, p, p], [2, 1, [2, 1, 3, 1]] ], [ [p, p, p], [1, [3, 1]], [2, 1, [2, 1, 3, 1]] ] ];
if Length(arg) = 0 then
list := List(P3, x -> SOTRec.groupFromData(x));
else
list := P3;
fi;
return list;
elif p = 2 and k = 3 then
order8 := [ [ [2, 2, 2], [1, [2, 1]], [2, [3, 1]] ], [ [2, 2, 2], [1, [2, 1]] ], [ [2, 2, 2] ], [ [2, 2, 2], [2, 1, [2, 1, 3, 1]] ], [ [2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ] ];
if Length(arg) = 0 then
list := List(order8, x -> SOTRec.groupFromData(x));
else
list := order8;
fi;
return list;
fi;
if p > 3 and k = 4 then
r := Int(Z(p));
P4 := [ [ [p, p, p, p], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ], [ [p, p, p, p], [1, [2, 1]], [2, [3, 1]] ], [ [p, p, p, p], [1, [2, 1]], [3, [4, 1]] ], [ [p, p, p, p], [1, [2, 1]] ], [ [p, p, p, p] ],
[ [p, p, p, p], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [1, [2, 1]], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [p, p, p, p], [1, [4, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [3, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, r, 4, 1]] ],
[ [p, p, p, p], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [1, [2, 1]], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ] ];
if Length(arg) = 0 then
list := List(P4, x -> SOTRec.groupFromData(x));
else
list := P4;
fi;
return list;
elif p = 3 and k = 4 then
order81 := [ [ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [3, [4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]] ],
[ [3, 3, 3, 3] ],
[ [3, 3, 3, 3], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [4, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [3, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 2, 4, 1]] ],
[ [3, 3, 3, 3], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 2, 4, 1]] ] ];
if Length(arg) = 0 then
list := List(order81, x -> SOTRec.groupFromData(x));
else
list := order81;
fi;
return list;
elif p = 2 and k = 4 then
order16 := [ [ [2, 2, 2, 2], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]], [2, [3, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]], [3, [4, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]] ],
[ [2, 2, 2, 2] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [2, 2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1,[2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1,[2, 1, 3, 1]], [3, 1,[3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 3, 1, 4, 1]], [3, 1, [3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [1, [4, 1]], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 3, 1, 4, 1]], [3, 1, [3, 1, 4, 1]] ] ];
if Length(arg) = 0 then
list := List(order16, x -> SOTRec.groupFromData(x));
else
list := order16;
fi;
return list;
elif k > 4 then
Error("AllSOTGroups is not available for p-groups of order not dividing p^4.");
elif p = 1 and k = 1 then
list := [AbelianGroup([1])];
return list;
fi;
end;
######################################
SOTRec.NumberPGroups := function(p, k)
if k = 1 then
return 1;
elif k = 2 then
return 2;
elif k = 3 then
return 5;
elif k = 4 and p = 2 then
return 14;
elif k = 4 and p > 2 then
return 15;
fi;
end;
#####################################
SOTRec.PGroup := function(p, k, i, arg...)
local PG, P2, P3, order8, P4, order16, order81, r, G;
if p > 1 then
if k = 1 then
PG := [ [p] ];
if i = 1 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(PG);
elif Length(arg) = 1 then
G := PG;
fi;
else
Error("There is a unique group of prime order up to isomorphism.");
fi;
fi;
if k = 2 then
P2 := [ [ [p, p], [1, [2, 1]] ], [ [p, p] ] ];
if i < 3 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(P2[i]);
elif Length(arg) = 1 then
G := P2[i];
fi;
else
Error("There are two groups of prime-squared order up to isomorphism: one is cyclic and the other elementary abelian.");
fi;
fi;
if p > 2 and k = 3 then
P3 := [ [ [p, p, p], [1, [2, 1]], [2, [3, 1]] ], [ [p, p, p], [1, [2, 1]] ], [ [p, p, p] ], [ [p, p, p], [2, 1, [2, 1, 3, 1]] ], [ [p, p, p], [1, [3, 1]], [2, 1, [2, 1, 3, 1]] ] ];
if i < 6 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(P3[i]);
elif Length(arg) = 1 then
G := P3[i];
fi;
else
Error("There are five isomorphism types of groups of prime-cubed order.");
fi;
elif p = 2 and k = 3 then
order8 := [ [ [2, 2, 2], [1, [2, 1]], [2, [3, 1]] ], [ [2, 2, 2], [1, [2, 1]] ], [ [2, 2, 2] ], [ [2, 2, 2], [2, 1, [2, 1, 3, 1]] ], [ [2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ] ];
if i < 6 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(order8[i]);
elif Length(arg) = 1 then
G := order8[i];
fi;
else
Error("There are five isomorphism types of groups of order 8.");
fi;
fi;
if p > 3 and k = 4 then
r := Int(Z(p));
P4 := [ [ [p, p, p, p], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ], [ [p, p, p, p], [1, [2, 1]], [2, [3, 1]] ], [ [p, p, p, p], [1, [2, 1]], [3, [4, 1]] ], [ [p, p, p, p], [1, [2, 1]] ], [ [p, p, p, p] ],
[ [p, p, p, p], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [1, [2, 1]], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [p, p, p, p], [1, [4, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [3, 1, [2, 1, 3, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, r, 4, 1]] ],
[ [p, p, p, p], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [p, p, p, p], [1, [2, 1]], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ] ];
if i < 16 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(P4[i]);
elif Length(arg) = 1 then
G := P4[i];
fi;
else
Error("There are 15 isomorphism types of groups of this order.");
fi;
elif p = 3 and k = 4 then
r := Int(Z(p));
order81 := [ [ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [3, [4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]] ], [ [3, 3, 3, 3] ],
[ [3, 3, 3, 3], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [2, 1]], [2, [3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [4, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [3, 1, [2, 1, 3, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, r, 4, 1]] ],
[ [3, 3, 3, 3], [3, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [3, 3, 3, 3], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 4, 1]], [4, 1, [3, r, 4, 1]] ] ];
if i < 16 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(order81[i]);
elif Length(arg) = 1 then
G := order81[i];
fi;
else
Error("There are 15 isomorphism types of groups of order 81.");
fi;
elif p = 2 and k = 4 then
order16 := [ [ [2, 2, 2, 2], [1, [2, 1]], [2, [3, 1]], [3, [4, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]], [2, [3, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]], [3, [4, 1]] ],
[ [2, 2, 2, 2], [1, [2, 1]] ],
[ [2, 2, 2, 2] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 3, 1]], [4, 1, [3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 3, 1]] ],
[ [2, 2, 2, 2], [1, [3, 1]], [2, [3, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 3, 1]], [3, 1, [3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 3, 1, 4, 1]], [3, 1, [3, 1, 4, 1]] ],
[ [2, 2, 2, 2], [1, [4, 1]], [2, [3, 1]], [3, [4, 1]], [2, 1, [2, 1, 3, 1, 4, 1]], [3, 1, [3, 1, 4, 1]] ] ];
if i < 15 then
if Length(arg) = 0 then
G := SOTRec.groupFromData(order16[i]);
elif Length(arg) = 1 then
G := order16[i];
fi;
else
Error("There are 14 isomorphism types of groups of order 16.");
fi;
fi;
if k > 4 then
Error("SOTGroup is not available for p-groups of order not dividing p^4.");
fi;
else
G := AbelianGroup([1]);
fi;
return G;
end;
[ Dauer der Verarbeitung: 0.29 Sekunden
(vorverarbeitet)
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