Quelle p2qr.gi
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Untersuchungsergebnis.gi Download desUnknown {[0] [0] [0]}zum Wurzelverzeichnis wechseln
## The following functions contribute to AllSOTGroups, NumberOfSOTGroups, and SOTGroup, respectively.
## Let G be a group of order p^2qr (p, q, r are distinct primes and q < r).
## Since G is cubefree, G is nilpotent if and only if G is abelian.
## By examining the composition series (with simple factors) of G, we observe that G is nonsolvable if and only if it is isomorphic to \Alt_5.
## If G is non-nilpotent and solvable, then it has a nontrivial, abelian Fitting subgroup, denoted by F.
## By [2, Lemma 3.7], we know G/F acts faithfully on F, that is, G/F embeds into Aut(F).
## We deduce that |F| \in \{r, qr, p^2, p^2q, p^2r, pr, pqr\}.
## For each case depending on the size of F, we construct the isomorphism types of non-nilpotent, solvable groups G.
## For further details, see [2, Section 3.2 & 3.7].
SOTRec.allGroupsP2QR := function(p, q, r)
local a, b, c, u, v, ii, qq, iii, qqq, k, l, Rootpr, Rootpq, Rootrq, Rootrp, Rootrp2, Rootqp, Rootqp2,
rootpr, rootpq, rootrq, rootrp, rootrp2, rootqp, rootqp2, matq, matr, matqr, mat, mat_k, all, list;
####
Assert(1, r > q);
Assert(1, IsPrimeInt(p));
Assert(1, IsPrimeInt(q));
Assert(1, IsPrimeInt(r));
a := Z(r); #\sigma_r
b := Z(p); #\sigma_p
c := Z(q); #\sigma_q
u := ZmodnZObj(Int(Z(p)), p^2);
if not u^(p-1) = ZmodnZObj(1, p^2) then
v := u;
else
v := u + 1;
fi;
if (p - 1) mod q = 0 then
ii := Int(v^((p^2-p)/q)) mod p;
qq := (Int(v^((p^2-p)/q)) - ii)/p;
fi;
if (p - 1) mod r = 0 then
iii := Int(v^((p^2-p)/r)) mod p;
qqq := (Int(v^((p^2-p)/r)) - iii)/p;
fi;
if (p + 1) mod (q * r) = 0 and q > 2 then
matqr := SOTRec.QthRootGL2P(p, (q*r));
mat := matqr^q;
fi;
if (p + 1) mod r = 0 and r > 2 then
matr := SOTRec.QthRootGL2P(p, r);
fi;
if (p + 1) mod q = 0 and q > 2 then
matq := SOTRec.QthRootGL2P(p, q);
fi;
############ abelian groups:
all := [ [ [p, p, q, r], [1, [2, 1]] ], [ [p, p, q, r] ] ];
############ precompute canonical roots:
if (r - 1) mod p = 0 then
rootrp := a^((r-1)/p);
Rootrp := Int(rootrp);
fi; #\rho(r, p)
if (r - 1) mod (p^2) = 0 then
rootrp2 := a^((r-1)/(p^2));
Rootrp2 := Int(rootrp2);
fi; #\rho(r, p^2)
if (r - 1) mod q = 0 then
rootrq := a^((r-1)/q);
Rootrq := Int(rootrq);
fi; #\rho(r, q)
if (p - 1) mod r = 0 then
rootpr := b^((p-1)/r);
Rootpr := Int(rootpr);
fi; #\rho(p, r)
if (p - 1) mod q = 0 then
rootpq := b^((p-1)/q);
Rootpq := Int(rootpq);
fi; #\rho(p, q)
if (q - 1) mod p = 0 then
rootqp := c^((q-1)/p);
Rootqp := Int(rootqp);
fi; #\rho(q, p)
if (q - 1) mod (p^2) = 0 then
rootqp2 := c^((q-1)/(p^2));
Rootqp2 := Int(rootqp2);
fi; #\rho(q, p^2)
############ case 1: nonabelian and Fitting subgroup has order r -- p^2q | (r - 1), unique up to isomorphism
if (r - 1) mod (p^2*q) = 0 then ##C_{p^2q} \ltimes C_r
Add(all, [ [p, p, q, r], [1, [2, 1]], [2, [3, 1]], [4, 1, [4, Int(a^((r-1)/(p^2*q)))]], [4, 2, [4, Int(a^((r-1)/(p*q)))]], [4, 3, [4, Rootrq]] ]);
fi;
############ case 2: nonabelian and Fitting subgroup has order qr -- p^2 | (q - 1)(r - 1) or p | (q - 1)(r - 1)
if (q - 1) mod (p^2) = 0 then ## p^2 | (q - 1) and G \cong (C_{p^2} \ltimes C_q) \times C_r
Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]] ]);
fi;
if (q - 1) mod (p^2) = 0 and (r - 1) mod p = 0 then ## p^2 | (q - 1), p | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
if (q - 1) mod (p^2) = 0 and (r - 1) mod (p^2) = 0 then ## p^2 | (q - 1), p^2 | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in SOTRec.groupofunitsP2(p) do
Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]], [4, 1, [4, Int(rootrp2^k)]], [4, 2, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod (p^2) = 0 then ## p^2 | (r - 1), and G \cong (C_{p^2} \ltimes C_r) \times C_q
Add(all, [ [p, p, q, r], [1, [2, 1]], [4, 1, [4, Rootrp2]], [4, 2, [4, Rootrp]] ]);
fi;
if (r - 1) mod (p^2) = 0 and (q - 1) mod p = 0 then ## p | (q - 1), p^2 | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1]
do Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp2^k)]], [4, 2, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## p | (q - 1), p | (r - 1), and G \cong (C_p \ltimes C_q) \times (C_p \ltimes C_r)
Add(all, [ [p, p, q, r], [3, 1, [3, Rootqp]], [4, 2, [4, Rootrp]] ]);
fi;
############ case 3: nonabelian and Fitting subgroup has order p^2 -- qr | (p^2 - 1)
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_{p^2}
Add(all, [ [q, r, p, p], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [3, 2, [3, iii, 4, qqq]], [4, 1, [4, ii]], [4, 2, [4, iii]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong (C_q \ltimes C_p) \times (C_r \ltimes C_p)
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [4, 2, [4, Rootpr]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong (C_{qr} \ltimes C_p) \times C_p
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [1..(q - 1)] do
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^k)]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [1..(r - 1)] do
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 2, [4, Int(rootpr^k)]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 and q > 2 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [0..(q - 1)/2] do
for l in [0..(r - 1)/2] do
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^(Int(c^k)))]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
od;
fi;
if (p - 1) mod (q*r) =0 and q > 2 then
for k in [1..(q-3)/2] do
for l in [(r+1)/2..(r-2)] do
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^(Int(c^k)))]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
od;
fi;
if (p - 1) mod (q*r) = 0 and q = 2 then
for l in [0..(r-1)/2] do
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Rootpq]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 and q = 2 then ##q = 2, r | (p - 1), and G \cong D_r \ltimes C_p^2
Add(all, [ [q, r, p, p], [2, 1, [2, r-1]], [3, 1, [4, 1]], [3, 2, [3, Rootpr]], [4, 1, [3, 1]], [4, 2, [4, Int(rootpr^(-1))]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q > 2 then ## qr | (p + 1), q > 2, and G \cong C_{qr} \ltimes C_p^2
Add(all, [ [q, r, p, p], [1, [2, 1]],
[3, 1, [3, Int(matqr[1][1]), 4, Int(matqr[2][1])]],
[4, 1, [3, Int(matqr[1][2]), 4, Int(matqr[2][2])]],
[3, 2, [3, Int(mat[1][1]), 4, Int(mat[2][1])]],
[4, 2, [3, Int(mat[1][2]), 4, Int(mat[2][2])]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q = 2 then ## qr | (p + 1), q = 2, and G \cong C_{qr} \ltimes C_p^2
Add(all, [ [q, r, p, p], [3, 1, [3, p - 1]], [4, 1, [4, p - 1]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q = 2 then ## qr | (p + 1), q = 2, and G \cong (C_q \ltimes C_r)\ltimes C_p^2
Add(all, [ [q, r, p, p], [2, 1, [2, r-1]], [3, 1, [4, 1]], [4, 1, [3, 1]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p + 1) mod r = 0 and (p - 1) mod q = 0 and q > 2 then ## q | (p - 1), r | (p + 1), and G \cong (C_q \times C_r) \ltimes C_p^2
Add(all, [ [q, r, p, p], [3, 1, [3, Rootpq]], [4, 1, [4, Rootpq]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p - 1) mod r = 0 and (p + 1) mod q = 0 and q > 2 then ## q | (p + 1), r | (p - 1), and G \cong (C_q \times C_r) \ltimes C_p^2
Add(all, [ [q, r, p, p],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]],
[3, 2, [3, Rootpr]], [4, 2, [4, Rootpr]] ]);
fi;
############ case 4: nonabelian and Fitting subgroup has order p^2q -- r | (p - 1) or r | (p + 1)
if (p - 1) mod r = 0 then ## r | (p - 1) and G \cong (C_r \ltimes C_{p^2}) \times C_q
Add(all, [ [r, p, p, q], [2, [3, 1]], [2, 1, [2, iii, 3, qqq]], [3, 1, [3, iii]] ]);
fi;
if (p - 1) mod r = 0 then ## r | (p - 1) and G \cong (C_r \ltimes C_p) \times (C_p \times C_q)
Add(all, [ [r, p, p, q], [2, 1, [2, Rootpr]] ]);
fi;
if (p - 1) mod r = 0 and r > 2 then ## r | (p - 1) and G \cong (C_r \ltimes C_p^2) \times C_q
for k in [0..(r - 1)/2] do
Add(all, [ [r, p, p, q], [2, 1, [2, Rootpr]], [3, 1, [3, Int(rootpr^(Int(a^k)))]] ]);
od;
fi;
if (p + 1) mod r = 0 and r > 2 then ## r | (p + 1) and G \cong (C_r \ltimes C_p^2) \times C_q
Add(all, [ [r, p, p, q],
[2, 1, [2, Int(matr[1][1]), 3, Int(matr[2][1])]],
[3, 1, [2, Int(matr[1][2]), 3, Int(matr[2][2])]] ]);
fi;
############ case 5: nonabelian and Fitting subgroup has order p^2r -- q | (p - 1)(r - 1) or q | (p + 1)(r - 1)
if (r - 1) mod q = 0 then ## q \mid (r - 1) and G \cong (C_q \ltimes C_r) \times C_{p^2}
Add(all, [ [q, r, p, p], [3, [4, 1]], [2, 1, [2, Rootrq]] ]);
fi;
if (p - 1) mod q = 0 then ## q \mid (p - 1) and G \cong (C_q \ltimes C_{p^2}) \times C_r
Add(all, [ [q, p, p, r], [2, [3, 1]], [2, 1, [2, ii, 3, qq]], [3, 1, [3, ii]] ]);
fi;
if (p - 1) mod q = 0 and (r - 1) mod q = 0 then ## q \mid (p - 1) and G \cong C_q \ltimes (C_{p^2} \times C_r)
for k in [1..(q-1)] do
Add(all, [ [q, p, p, r], [2, [3, 1]], [2, 1, [2, ii, 3, qq]], [3, 1, [3, ii]], [4, 1, [4, Int(rootrq^k)]] ]);
od;
fi;
if (p - 1) mod q = 0 then ## q \mid (p - 1) and G \cong (C_q \ltimes C_p) \times C_p \times C_r
Add(all, [ [q, p, p, r], [2, 1, [2, Rootpq]] ]);
fi;
if (p - 1) mod q = 0 and q > 2 then ## q | (p - 1) and G \cong (C_q \ltimes C_p^2) \times C_r
for k in [0..(q - 1)/2] do
Add(all, [ [q, p, p, r], [2, 1, [2, Rootpq]], [3, 1, [3, Int(rootpq^(Int(c^k)))]] ]);
od;
fi;
if (p - 1) mod q = 0 and q = 2 then
Add(all, [ [q, p, p, r], [2, 1, [2, p - 1]], [3, 1, [3, p - 1]] ]);
fi;
if (p + 1) mod q = 0 and q > 2 then ## q | (p + 1), and G \cong (C_q \ltimes C_p^2) \times C_r
Add(all, [ [q, r, p, p], [3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]], [4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (r - 1) mod q = 0 then ## q | (r - 1), and G \cong (C_q \ltimes C_r) \times C_p^2
Add(all, [ [q, r, p, p], [2, 1, [2, Rootrq]] ]);
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 then ## q | (r - 1), q | (p - 1), and G \cong C_q \ltimes (C_r \times C_p) \times C_p
for k in [1..q - 1] do
Add(all, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Int(rootpq^k)]] ]);
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then ## q | (r - 1), q | (p - 1), and G \cong C_q \ltimes (C_r \times C_p^2)
for k in [0..(q - 3)/2] do
Add(all, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
for l in [1..(q - 3)/2] do
for k in [0..(q - 1)/2] do
Add(all, [ [q, r, p, p], [2, 1, [2, Int(rootrq^(Int(c^l)))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
Add(all, [ [q, r, p, p], [2, 1, [2, Int(a^(Int(c^((q - 1)/2))*(r-1)/q))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(b^(Int(c^((q - 1)/2))*(p-1)/q))]] ]);
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
for l in [(q - 1)/2..q - 2] do
for k in [0..(q - 3)/2] do
Add(all, [ [q, r, p, p], [2, 1, [2, Int(rootrq^(Int(c^l)))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
od;
fi;
if q = 2 then
Add(all, [ [q, r, p, p], [2, 1, [2, r - 1]], [3, 1, [3, Rootpq]], [4, 1, [4, Rootpq]] ]);
fi;
if (p + 1) mod q = 0 and (r - 1) mod q = 0 and q > 2 then ## q | (r - 1), q | (p + 1), and G \cong C_q \ltimes (C_r \times C_p^2)
for k in [1..(q-1)/2] do
mat_k := matq^k;
Add(all, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Int(mat_k[1][1]), 4, Int(mat_k[2][1])]], [4, 1, [3, Int(mat_k[1][2]), 4, Int(mat_k[2][2])]] ]);
od;
fi;
############ case 6: nonabelian and Fitting subgroup has order pr -- q | (p - 1)(r - 1) and p | (r - 1)
if (r - 1) mod p = 0 and (r - 1) mod q = 0 then ## q | (r - 1), p | (r - 1), and G \cong (C_p \times C_q) \ltimes C_r \times C_p
Add(all, [ [p, q, p, r], [4, 1, [4, Rootrp]], [4, 2, [4, Rootrq]] ]);
fi;
if (r - 1) mod p = 0 and (r - 1) mod q = 0 then ## q | (r - 1), p | (r - 1), and G \cong (C_p \times C_q) \ltimes C_r \times C_p
Add(all, [ [p, q, p, r], [1, [3, 1]], [4, 1, [4, Rootrp]], [4, 2, [4, Rootrq]] ]);
fi;
if (r - 1) mod p = 0 and (p - 1) mod q = 0 then ## q | (p - 1), p | (r - 1), and G \cong (C_p \ltimes C_r) \times (C_q \ltimes C_p)
Add(all, [ [p, r, q, p], [2, 1, [2, Rootrp]], [4, 3, [4, Rootpq]] ]);
fi;
if (r - 1) mod p = 0 and (p - 1) mod q = 0 and (r - 1) mod q = 0 then ## q | (p - 1), p | (r - 1), q | (r - 1), and G \cong (C_p \times C_q) \ltimes (C_r \times C_p)
for k in [1..q-1] do
Add(all, [ [p, q, r, p], [3, 1, [3, Rootrp]], [3, 2, [3, Rootrq]], [4, 2, [4, Int(rootpq^k)]] ]);
od;
fi;
############ case 7: nonabelian and Fitting subgroup has order pqr -- p | (r - 1)(q - 1)
if (r - 1) mod p = 0 then ## P \cong C_{p^2}, p | (r - 1) and G \cong (C_{p^2} \ltimes C_r) \times C_q
Add(all, [ [p, p, r, q], [1, [2, 1]], [3, 1, [3, Rootrp]] ]);
fi;
if (q - 1) mod p = 0 then ## P \cong C_{p^2}, p | (q - 1) and G \cong (C_{p^2} \ltimes C_q) \times C_r
Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]] ]);
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## P \cong C_{p^2} and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(all, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod p = 0 then ## P \cong C_p^2, p | (r - 1) and G \cong C_p \times (C_p \ltimes C_r) \times C_q
Add(all, [ [p, p, q, r], [4, 1, [4, Rootrp]] ]);
fi;
if (q - 1) mod p = 0 then ## P \cong C_p^2, p | (r - 1) and G \cong C_p \times (C_p \ltimes C_r) \times C_r
Add(all, [ [p, p, q, r], [3, 1, [3, Rootqp]] ]);
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## P \cong C_p^2 and G \cong C_p^2 \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(all, [ [p, p, q, r], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
############
list := List(all, x->SOTRec.groupFromData(x));
if p = 2 and q = 3 and r = 5 then
Add(list, AlternatingGroup(5));
fi;
return list;
end;
######################################################
SOTRec.NumberGroupsP2QR := function(p, q, r)
local m;
if p = 2 and q = 3 and r = 5 then
m := 13;
elif q = 2 then
m := 10 + (2*r + 7)*SOTRec.w((p - 1), r)
+ 3*SOTRec.w((p + 1), r)
+ 6*SOTRec.w((r - 1), p)
+ 2*SOTRec.w((r - 1), (p^2));
elif q > 2 and not (p = 2 and q = 3 and r = 5) then
m := 2 + (p^2 - p)*SOTRec.w((q - 1), (p^2))*SOTRec.w((r - 1), (p^2))
+ (p - 1)*(SOTRec.w((q - 1), (p^2))*SOTRec.w((r - 1), p) + SOTRec.w((r - 1), (p^2))*SOTRec.w((q - 1), p)
+ 2*SOTRec.w((r - 1), p)*SOTRec.w((q - 1), p))
+ (q - 1)*(q + 4)*SOTRec.w((p - 1), q)*SOTRec.w((r - 1), q)/2
+ (q - 1)*(SOTRec.w((p + 1), q)*SOTRec.w((r - 1), q)
+ SOTRec.w((p - 1), q)
+ SOTRec.w((p - 1), (q*r))
+ 2*SOTRec.w((r - 1), (p*q))*SOTRec.w((p - 1), q))/2
+ (q*r + 1)*SOTRec.w((p - 1), (q*r))/2
+ (r + 5)*SOTRec.w((p - 1), r)*(1 + SOTRec.w((p - 1), q))/2
+ SOTRec.w((p^2 - 1), (q*r))
+ 2*SOTRec.w((r - 1), (p*q))
+ SOTRec.w((r - 1), p)*SOTRec.w((p - 1), q)
+ SOTRec.w((r - 1), (p^2*q))
+ SOTRec.w((r - 1), p)*SOTRec.w((q - 1), p)
+ 2*SOTRec.w((q - 1), p)
+ 3*SOTRec.w((p - 1), q)
+ 2*SOTRec.w((r - 1), p)
+ 2*SOTRec.w((r - 1), q)
+ SOTRec.w((r - 1), (p^2))
+ SOTRec.w((q - 1), p^2)
+ SOTRec.w((p + 1), r)
+ SOTRec.w((p + 1), q);
fi;
return m;
end;
######################################################
######################################################
SOTRec.GroupP2QR := function(p, q, r, i)
local a, b, c, u, v, ii, qq, iii, qqq, k, l, matq, matr, matqr, mat, mat_k,
Rootpr, Rootpq, Rootrq, Rootrp, Rootrp2, Rootqp, Rootqp2, rootpr, rootpq, rootrq, rootrp, rootrp2, rootqp, rootqp2,
c1, c2, c3, c4, c5, c6, c7, l1, l2, l3, l4, l5, l6, l7, l0, data, G;
####
Assert(1, r > q);
Assert(1, IsPrimeInt(p));
Assert(1, IsPrimeInt(q));
Assert(1, IsPrimeInt(r));
############ precompute Roots:
a := Z(r);
b := Z(p);
c := Z(q);
if (r - 1) mod p = 0 then
rootrp := a^((r-1)/p);
Rootrp := Int(rootrp);
fi;
if (r - 1) mod (p^2) = 0 then
rootrp2 := a^((r-1)/(p^2));
Rootrp2 := Int(rootrp2);
fi;
if (r - 1) mod q = 0 then
rootrq := a^((r-1)/q);
Rootrq := Int(rootrq);
fi;
if (p - 1) mod r = 0 then
rootpr := b^((p-1)/r);
Rootpr := Int(rootpr);
fi;
if (p - 1) mod q = 0 then
rootpq := b^((p-1)/q);
Rootpq := Int(rootpq);
fi;
if (q - 1) mod p = 0 then
rootqp := c^((q-1)/p);
Rootqp := Int(rootqp);
fi;
if (q - 1) mod (p^2) = 0 then
rootqp2 := c^((q-1)/(p^2));
Rootqp2 := Int(rootqp2);
fi;
if not Int(b)^(p-1) mod p^2 = 1 then
v := ZmodnZObj(Int(b), p^2);
else
v := ZmodnZObj(Int(b) + p, p^2);
fi;
if (p - 1) mod q = 0 then
ii := Int(v^((p^2-p)/q)) mod p;
qq := (Int(v^((p^2-p)/q)) - ii)/p;
fi;
if (p - 1) mod r = 0 then
iii := Int(v^((p^2-p)/r)) mod p;
qqq := (Int(v^((p^2-p)/r)) - iii)/p;
fi;
if (p + 1) mod (q * r) = 0 and q > 2 then
matqr := SOTRec.QthRootGL2P(p, (q*r));
mat := matqr^q;
fi;
if (p + 1) mod r = 0 and r > 2 then
matr := SOTRec.QthRootGL2P(p, r);
fi;
if (p + 1) mod q = 0 and q > 2 then
matq := SOTRec.QthRootGL2P(p, q);
fi;
############ enumeration of distinct cases
c1 := SOTRec.w((r - 1), p^2*q);;
c2 := SOTRec.w((q - 1), p^2) + (p - 1)*SOTRec.w((q - 1), p^2)*SOTRec.w((r - 1), p)
+ (p^2 - p)*SOTRec.w((r - 1), p^2)*SOTRec.w((q - 1), p^2)
+ SOTRec.w((r - 1), p^2) + (p - 1)*SOTRec.w((q - 1), p)*SOTRec.w((r - 1), p^2)
+ SOTRec.w((r - 1), p)*SOTRec.w((q - 1), p);;
c3 := 1/2*(q*r+q+r+7)*SOTRec.w((p - 1), q*r)
+ SOTRec.w((p^2 - 1), q*r)*(1 - SOTRec.w((p - 1), q*r))*(1 - SOTRec.delta(q, 2))
+ 2*SOTRec.w((p + 1), r)*SOTRec.delta(q, 2);;
c4 := 1/2*(r + 5)*SOTRec.w((p - 1), r) + SOTRec.w((p + 1), r);;
c5 := 8*SOTRec.delta(q, 2)
+ (1 - SOTRec.delta(q, 2))*(1/2*(q - 1)*(q + 4)*SOTRec.w((p - 1), q)*SOTRec.w((r - 1), q)
+ 1/2*(q - 1)*SOTRec.w((p + 1), q)*SOTRec.w((r - 1), q)
+ 1/2*(q + 5)*SOTRec.w((p - 1), q)
+ 2*SOTRec.w((r - 1), q)
+ SOTRec.w((p + 1), q));;
c6 := SOTRec.w((r - 1), p)*(SOTRec.w((p - 1), q)*(1 + (q - 1)*SOTRec.w((r - 1), q))
+ 2*SOTRec.w((r - 1), q));;
c7 := 2*(SOTRec.w((q - 1), p) + SOTRec.w((r - 1), p) +
(p - 1)*SOTRec.w((q - 1), p)*SOTRec.w((r - 1), p));
############ add abelian groups in:
l0 := [ [ [p, p, q, r], [1, [2, 1]] ], [ [p, p, q, r] ] ];
if i < 3 then
G := SOTRec.groupFromData(l0[i]);
return G;
fi;
############ case 1: nonabelian and Fitting subgroup has order r -- unique isomorphism type iff p^2q | (r - 1)
if (r - 1) mod (p^2*q) = 0 and i > 2 and i < (3 + c1) then ##C_{p^2q} \ltimes C_r
l1 := [];
Add(l1, [ [p, p, q, r], [1, [2, 1]], [2, [3, 1]], [4, 1, [4, Int(a^((r-1)/(p^2*q)))]], [4, 2, [4, Int(a^((r-1)/(p*q)))]], [4, 3, [4, Rootrq]] ]);
data := l1[i - 2];
return SOTRec.groupFromData(data);
fi;
############ case 2: nonabelian and Fitting subgroup has order qr -- p^2 | (q - 1)(r - 1) or (p^2 | (q - 1) or (r - 1)) or (p | (q - 1) and p | (r - 1))
if i > (2 + c1) and i < (3 + c1 + c2) then
l2 := [];
if (q - 1) mod (p^2) = 0 then ## p^2 | (q - 1) and G \cong (C_{p^2} \ltimes C_q) \times C_r
Add(l2, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]] ]);
fi;
if (q - 1) mod (p^2) = 0 and (r - 1) mod p = 0 then ## p^2 | (q - 1), p | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(l2, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
if (q - 1) mod (p^2) = 0 and (r - 1) mod (p^2) = 0 then ## p^2 | (q - 1), p^2 | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in SOTRec.groupofunitsP2(p) do
Add(l2, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp2]], [3, 2, [3, Rootqp]], [4, 1, [4, Int(rootrp2^k)]], [4, 2, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod (p^2) = 0 then ## p^2 | (r - 1), and G \cong (C_{p^2} \ltimes C_r) \times C_q
Add(l2, [ [p, p, q, r], [1, [2, 1]], [4, 1, [4, Rootrp2]], [4, 2, [4, Rootrp]] ]);
fi;
if (r - 1) mod (p^2) = 0 and (q - 1) mod p = 0 then ## p | (q - 1), p^2 | (r - 1), and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1]
do Add(l2, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp2^k)]], [4, 2, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## p | (q - 1), p | (r - 1), and G \cong (C_p \ltimes C_q) \times (C_p \ltimes C_r)
Add(l2, [ [p, p, q, r], [3, 1, [3, Rootqp]], [4, 2, [4, Rootrp]] ]);
fi;
data := l2[i - 2 - c1];
return SOTRec.groupFromData(data);
fi;
############ case 3: nonabelian and Fitting subgroup has order p^2 -- qr | (p^2 - 1)
if i > (2 + c1 + c2) and i < (3 + c1 + c2 + c3) then
l3 := [];
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_{p^2}
Add(l3, [ [q, r, p, p], [3, [4, 1]], [3, 1, [3, ii, 4, qq]], [3, 2, [3, iii, 4, qqq]], [4, 1, [4, ii]], [4, 2, [4, iii]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong (C_q \ltimes C_p) \times (C_r \ltimes C_p)
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [4, 2, [4, Rootpr]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong (C_{qr} \ltimes C_p) \times C_p
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]] ]);
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [1..(q - 1)] do
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^k)]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [1..(r - 1)] do
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 2, [4, Int(rootpr^k)]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 and q > 2 then ## qr | (p - 1) and G \cong C_{qr} \ltimes C_p^2
for k in [0..(q - 1)/2] do
for l in [0..(r - 1)/2] do
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^(Int(c^k)))]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
od;
fi;
if (p - 1) mod (q*r) =0 and q > 2 then
for k in [1..(q-3)/2] do
for l in [(r+1)/2..(r-2)] do
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Int(rootpq^(Int(c^k)))]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
od;
fi;
if (p - 1) mod (q*r) = 0 and q = 2 then
for l in [0..(r-1)/2] do
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [3, 2, [3, Rootpr]], [4, 1, [4, Rootpq]], [4, 2, [4, Int(b^(Int(a^l)*(p-1)/r))]] ]);
od;
fi;
if (p - 1) mod (q*r) = 0 and q = 2 then ##q = 2, r | (p - 1), and G \cong (C_2 \ltimes C_r) \ltimes C_p^2
Add(l3, [ [q, r, p, p], [2, 1, [2, r-1]], [3, 1, [4, 1]], [3, 2, [3, Rootpr]], [4, 1, [3, 1]], [4, 2, [4, Int(b^((1 - p)/r))]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q > 2 then ## qr | (p + 1), q > 2, and G \cong C_{qr} \ltimes C_p^2
Add(l3, [ [q, r, p, p], [1, [2, 1]],
[3, 1, [3, Int(matqr[1][1]), 4, Int(matqr[2][1])]],
[4, 1, [3, Int(matqr[1][2]), 4, Int(matqr[2][2])]],
[3, 2, [3, Int(mat[1][1]), 4, Int(mat[2][1])]],
[4, 2, [3, Int(mat[1][2]), 4, Int(mat[2][2])]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q = 2 then ## qr | (p + 1), q = 2, and G \cong C_{qr} \ltimes C_p^2
Add(l3, [ [q, r, p, p], [3, 1, [3, p - 1]], [4, 1, [4, p - 1]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p + 1) mod (q*r) = 0 and q = 2 then ## qr | (p + 1), q = 2, and G \cong (C_q \ltimes C_r)\ltimes C_p^2
Add(l3, [ [q, r, p, p], [2, 1, [2, r-1]], [3, 1, [4, 1]], [4, 1, [3, 1]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p + 1) mod r = 0 and (p - 1) mod q = 0 and q > 2 then ## q | (p - 1), r | (p + 1), and G \cong (C_q \times C_r) \ltimes C_p^2
Add(l3, [ [q, r, p, p], [3, 1, [3, Rootpq]], [4, 1, [4, Rootpq]],
[3, 2, [3, Int(matr[1][1]), 4, Int(matr[2][1])]],
[4, 2, [3, Int(matr[1][2]), 4, Int(matr[2][2])]] ]);
fi;
if (p - 1) mod r = 0 and (p + 1) mod q = 0 and q > 2 then ## q | (p + 1), r | (p - 1), and G \cong (C_q \times C_r) \ltimes C_p^2
Add(l3, [ [q, r, p, p],
[3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]],
[4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]],
[3, 2, [3, Rootpr]], [4, 2, [4, Rootpr]] ]);
fi;
data := l3[i - 2 - c1 - c2];
return SOTRec.groupFromData(data);
fi;
############ case 4: nonabelian and Fitting subgroup has order p^2q -- r | (p - 1) or r | (p + 1)
if i > (2 + c1 + c2 + c3) and i < (3 + c1 + c2 + c3 + c4) then
l4 := [];
if (p - 1) mod r = 0 then ## r | (p - 1) and G \cong (C_r \ltimes C_{p^2}) \times C_q
Add(l4, [ [r, p, p, q], [2, [3, 1]], [2, 1, [2, iii, 3, qqq]], [3, 1, [3, iii]] ]);
fi;
if (p - 1) mod r = 0 then ## r | (p - 1) and G \cong (C_r \ltimes C_p) \times (C_p \times C_q)
Add(l4, [ [r, p, p, q], [2, 1, [2, Rootpr]] ]);
fi;
if (p - 1) mod r = 0 and r > 2 then ## r | (p - 1) and G \cong (C_r \ltimes C_p^2) \times C_q
for k in [0..(r - 1)/2] do
Add(l4, [ [r, p, p, q], [2, 1, [2, Rootpr]], [3, 1, [3, Int(rootpr^(Int(a^k)))]] ]);
od;
fi;
if (p + 1) mod r = 0 and r > 2 then ## r | (p + 1) and G \cong (C_r \ltimes C_p^2) \times C_q
Add(l4, [ [r, p, p, q],
[2, 1, [2, Int(matr[1][1]), 3, Int(matr[2][1])]],
[3, 1, [2, Int(matr[1][2]), 3, Int(matr[2][2])]] ]);
fi;
data := l4[i - 2 - c1 - c2 - c3];
return SOTRec.groupFromData(data);
fi;
############ case 5: nonabelian and Fitting subgroup has order p^2r -- q | (p - 1)(r - 1) or q | (p + 1)(r - 1)
if i > (2 + c1 + c2 + c3 + c4) and i < (3 + c1 + c2 + c3 + c4 + c5) then
l5 := [];
if (r - 1) mod q = 0 then ## q \mid (r - 1) and G \cong (C_q \ltimes C_r) \times C_{p^2}
Add(l5, [ [q, r, p, p], [3, [4, 1]], [2, 1, [2, Rootrq]] ]);
fi;
if (p - 1) mod q = 0 then ## q \mid (p - 1) and G \cong (C_q \ltimes C_{p^2}) \times C_r
Add(l5, [ [q, p, p, r], [2, [3, 1]], [2, 1, [2, ii, 3, qq]], [3, 1, [3, ii]] ]);
fi;
if (p - 1) mod q = 0 and (r - 1) mod q = 0 then ## q \mid (p - 1) and G \cong C_q \ltimes (C_{p^2} \times C_r)
for k in [1..(q-1)] do
Add(l5, [ [q, p, p, r], [2, [3, 1]], [2, 1, [2, ii, 3, qq]], [3, 1, [3, ii]], [4, 1, [4, Int(rootrq^k)]] ]);
od;
fi;
if (p - 1) mod q = 0 then ## q \mid (p - 1) and G \cong (C_q \ltimes C_p) \times C_p \times C_r
Add(l5, [ [q, p, p, r], [2, 1, [2, Rootpq]] ]);
fi;
if (p - 1) mod q = 0 and q > 2 then ## q | (p - 1) and G \cong (C_q \ltimes C_p^2) \times C_r
for k in [0..(q - 1)/2] do
Add(l5, [ [q, p, p, r], [2, 1, [2, Rootpq]], [3, 1, [3, Int(rootpq^(Int(c^k)))]] ]);
od;
fi;
if (p - 1) mod q = 0 and q = 2 then
Add(l5, [ [q, p, p, r], [2, 1, [2, p - 1]], [3, 1, [3, p - 1]] ]);
fi;
if (p + 1) mod q = 0 and q > 2 then ## q | (p + 1), and G \cong (C_q \ltimes C_p^2) \times C_r
Add(l5, [ [q, r, p, p], [3, 1, [3, Int(matq[1][1]), 4, Int(matq[2][1])]], [4, 1, [3, Int(matq[1][2]), 4, Int(matq[2][2])]] ]);
fi;
if (r - 1) mod q = 0 then ## q | (r - 1), and G \cong (C_q \ltimes C_r) \times C_p^2
Add(l5, [ [q, r, p, p], [2, 1, [2, Rootrq]] ]);
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 then ## q | (r - 1), q | (p - 1), and G \cong C_q \ltimes (C_r \times C_p) \times C_p
for k in [1..q-1] do
Add(l5, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Int(rootpq^k)]] ]);
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then ## q | (r - 1), q | (p - 1), and G \cong C_q \ltimes (C_r \times C_p^2)
for k in [0..(q - 3)/2] do
Add(l5, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
for l in [1..(q - 3)/2] do
for k in [0..(q - 1)/2] do
Add(l5, [ [q, r, p, p], [2, 1, [2, Int(rootrq^(Int(c^l)))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
od;
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
Add(l5, [ [q, r, p, p], [2, 1, [2, Int(a^((1-r)/q))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(b^((1-p)/q))]] ]);
fi;
if (r - 1) mod q = 0 and (p - 1) mod q = 0 and q > 2 then
for l in [(q - 1)/2..q - 2] do
for k in [0..(q - 3)/2] do
Add(l5, [ [q, r, p, p], [2, 1, [2, Int(rootrq^(Int(c^l)))]], [3, 1, [3, Rootpq]], [4, 1, [4, Int(rootpq^(Int(c^k)))]] ]);
od;
od;
fi;
if q = 2 then ##C_q \ltimes (C_r \times C_p^2)
Add(l5, [ [q, r, p, p], [2, 1, [2, r - 1]], [3, 1, [3, p - 1]], [4, 1, [4, p - 1]] ]);
fi;
if (p + 1) mod q = 0 and (r - 1) mod q = 0 and q > 2 then ## q | (r - 1), q | (p + 1), and G \cong C_q \ltimes (C_r \times C_p^2)
for k in [1..(q - 1)/2] do
mat_k := matq^k;
Add(l5, [ [q, r, p, p], [2, 1, [2, Rootrq]], [3, 1, [3, Int(mat_k[1][1]), 4, Int(mat_k[2][1])]], [4, 1, [3, Int(mat_k[1][2]), 4, Int(mat_k[2][2])]] ]);
od;
fi;
data := l5[i - 2 - c1 - c2 - c3 - c4];
return SOTRec.groupFromData(data);
fi;
############ case 6: nonabelian and Fitting subgroup has order pr -- q | (p - 1)(r - 1) and p | (r - 1)
if i > (2 + c1 + c2 + c3 + c4 + c5) and i < (3 + c1 + c2 + c3 + c4 + c5 + c6) then
l6 := [];
if (r - 1) mod p = 0 and (r - 1) mod q = 0 then ## q | (r - 1), p | (r - 1), and G \cong (C_p \times C_q) \ltimes C_r \times C_p
Add(l6, [ [p, q, p, r], [4, 1, [4, Rootrp]], [4, 2, [4, Rootrq]] ]);
fi;
if (r - 1) mod p = 0 and (r - 1) mod q = 0 then ## q | (r - 1), p | (r - 1), and G \cong (C_p \times C_q) \ltimes C_r \times C_p
Add(l6, [ [p, q, p, r], [1, [3, 1]], [4, 1, [4, Rootrp]], [4, 2, [4, Rootrq]] ]);
fi;
if (r - 1) mod p = 0 and (p - 1) mod q = 0 then ## q | (p - 1), p | (r - 1), and G \cong (C_p \ltimes C_r) \times (C_q \ltimes C_p)
Add(l6, [ [p, r, q, p], [2, 1, [2, Rootrp]], [4, 3, [4, Rootpq]] ]);
fi;
if (r - 1) mod p = 0 and (p - 1) mod q = 0 and (r - 1) mod q = 0 then ## q | (p - 1), p | (r - 1), q | (r - 1), and G \cong (C_p \times C_q) \ltimes (C_r \times C_p)
for k in [1..q-1] do
Add(l6, [ [p, q, r, p], [3, 1, [3, Rootrp]], [3, 2, [3, Rootrq]], [4, 2, [4, Int(rootpq^k)]] ]);
od;
fi;
data := l6[i - 2 - c1 - c2 - c3 - c4 - c5];
return SOTRec.groupFromData(data);
fi;
############ case 7: nonabelian and Fitting subgroup has order pqr -- p | (r - 1)(q - 1)
if i > (2 + c1 + c2 + c3 + c4 + c5 + c6) and i < (3 + c1 + c2 + c3 + c4 + c5 + c6 + c7) then
l7 := [];
if (r - 1) mod p = 0 then ## P \cong C_{p^2}, p | (r - 1) and G \cong (C_{p^2} \ltimes C_r) \times C_q
Add(l7, [ [p, p, r, q], [1, [2, 1]], [3, 1, [3, Rootrp]] ]);
fi;
if (q - 1) mod p = 0 then ## P \cong C_{p^2}, p | (q - 1) and G \cong (C_{p^2} \ltimes C_q) \times C_r
Add(l7, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]] ]);
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## P \cong C_{p^2} and G \cong C_{p^2} \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(l7, [ [p, p, q, r], [1, [2, 1]], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
if (r - 1) mod p = 0 then ## P \cong C_p^2, p | (r - 1) and G \cong C_p \times (C_p \ltimes C_r) \times C_q
Add(l7, [ [p, p, q, r], [4, 1, [4, Rootrp]] ]);
fi;
if (q - 1) mod p = 0 then ## P \cong C_p^2, p | (q - 1) and G \cong C_p \times (C_p \ltimes C_q) \times C_r
Add(l7, [ [p, p, q, r], [3, 1, [3, Rootqp]] ]);
fi;
if (r - 1) mod p = 0 and (q - 1) mod p = 0 then ## P \cong C_p^2 and G \cong C_p^2 \ltimes (C_q \times C_r)
for k in [1..p-1] do
Add(l7, [ [p, p, q, r], [3, 1, [3, Rootqp]], [4, 1, [4, Int(rootrp^k)]] ]);
od;
fi;
data := l7[i - 2 - c1 - c2 - c3 - c4 - c5 - c6];
return SOTRec.groupFromData(data);
fi;
############
if p = 2 and q = 3 and r = 5 and i = 13 then
return AlternatingGroup(5);
fi;
############
end;
[ Dauer der Verarbeitung: 0.97 Sekunden
]
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2026-03-28
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