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|
/**************************************************************************
* *
* Regina - A Normal Surface Theory Calculator *
* Computational Engine *
* *
* Copyright (c) 1999-2025, Ben Burton *
* For further details contact Ben Burton (bab@debian.org). *
* *
* This program is free software; you can redistribute it and/or *
* modify it under the terms of the GNU General Public License as *
* published by the Free Software Foundation; either version 2 of the *
* License, or (at your option) any later version. *
* *
* As an exception, when this program is distributed through (i) the *
* App Store by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or *
* (iii) Google Play by Google Inc., then that store may impose any *
* digital rights management, device limits and/or redistribution *
* restrictions that are required by its terms of service. *
* *
* This program is distributed in the hope that it will be useful, but *
* WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU *
* General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <https://www.gnu.org/licenses/>. *
* *
**************************************************************************/
#include "maths/permgroup.h"
namespace regina {
template <int n, bool cached>
typename PermGroup<n, cached>::iterator&
PermGroup<n, cached>::iterator::operator ++() {
int k = 1;
// Work out which pos_[k] needs to be incremented.
while (k < n && pos_[k] == group_->count_[k] - 1)
++k;
if (k == n) {
// Out of options.
pos_[0] = 1; // past-the-end
return *this;
}
// Conveniently, all the terms term_[i][j] that we _were_ using for i < k
// were identities, since we insist that term_[i][i] == id.
// Therefore the only term that we need to remove before the increment
// is the term for k.
if constexpr (cached)
current_ = current_.cachedComp(
group_->term_[group_->usable_[k][pos_[k]]][k]); /* inverse term */
else
current_ = current_ *
group_->term_[group_->usable_[k][pos_[k]]][k]; /* inverse term */
++pos_[k];
if constexpr (cached)
current_ = current_.cachedComp(
group_->term_[k][group_->usable_[k][pos_[k]]]);
else
current_ = current_ * group_->term_[k][group_->usable_[k][pos_[k]]];
if (k > 1) {
std::fill(pos_ + 1, pos_ + k, 0);
if constexpr (cached)
current_ = current_.cachedComp(group_->initSeq_[k - 1]);
else
current_ = current_ * group_->initSeq_[k - 1];
}
return *this;
}
template PermGroup<2, false>::iterator&
PermGroup<2, false>::iterator::operator ++();
template PermGroup<3, false>::iterator&
PermGroup<3, false>::iterator::operator ++();
template PermGroup<4, false>::iterator&
PermGroup<4, false>::iterator::operator ++();
template PermGroup<5, false>::iterator&
PermGroup<5, false>::iterator::operator ++();
template PermGroup<6, false>::iterator&
PermGroup<6, false>::iterator::operator ++();
template PermGroup<7, false>::iterator&
PermGroup<7, false>::iterator::operator ++();
template PermGroup<8, false>::iterator&
PermGroup<8, false>::iterator::operator ++();
template PermGroup<9, false>::iterator&
PermGroup<9, false>::iterator::operator ++();
template PermGroup<10, false>::iterator&
PermGroup<10, false>::iterator::operator ++();
template PermGroup<11, false>::iterator&
PermGroup<11, false>::iterator::operator ++();
template PermGroup<12, false>::iterator&
PermGroup<12, false>::iterator::operator ++();
template PermGroup<13, false>::iterator&
PermGroup<13, false>::iterator::operator ++();
template PermGroup<14, false>::iterator&
PermGroup<14, false>::iterator::operator ++();
template PermGroup<15, false>::iterator&
PermGroup<15, false>::iterator::operator ++();
template PermGroup<16, false>::iterator&
PermGroup<16, false>::iterator::operator ++();
template PermGroup<2, true>::iterator&
PermGroup<2, true>::iterator::operator ++();
template PermGroup<3, true>::iterator&
PermGroup<3, true>::iterator::operator ++();
template PermGroup<4, true>::iterator&
PermGroup<4, true>::iterator::operator ++();
template PermGroup<5, true>::iterator&
PermGroup<5, true>::iterator::operator ++();
template PermGroup<6, true>::iterator&
PermGroup<6, true>::iterator::operator ++();
template PermGroup<7, true>::iterator&
PermGroup<7, true>::iterator::operator ++();
template PermGroup<8, true>::iterator&
PermGroup<8, true>::iterator::operator ++();
template PermGroup<9, true>::iterator&
PermGroup<9, true>::iterator::operator ++();
template PermGroup<10, true>::iterator&
PermGroup<10, true>::iterator::operator ++();
template PermGroup<11, true>::iterator&
PermGroup<11, true>::iterator::operator ++();
template PermGroup<12, true>::iterator&
PermGroup<12, true>::iterator::operator ++();
template PermGroup<13, true>::iterator&
PermGroup<13, true>::iterator::operator ++();
template PermGroup<14, true>::iterator&
PermGroup<14, true>::iterator::operator ++();
template PermGroup<15, true>::iterator&
PermGroup<15, true>::iterator::operator ++();
template PermGroup<16, true>::iterator&
PermGroup<16, true>::iterator::operator ++();
template <int n, bool cached>
PermGroup<n, cached>::PermGroup(NamedPermGroup group) {
// Remember: all permutations not explicitly set here will be
// initialised to the identity.
switch (group) {
case NamedPermGroup::Symmetric:
for (int k = 1; k < n; ++k)
for (int j = 0; j < k; ++j) {
// These terms are all self-inverse.
term_[k][j] = term_[j][k] = Perm<n>(j, k);
}
for (int i = 0; i < n; ++i)
count_[i] = i + 1;
// Each usable_[i] should be the identity.
break;
case NamedPermGroup::Alternating:
for (int k = 2; k < n; ++k) {
// Each non-trivial term should be a 3-cycle.
if constexpr (cached) {
term_[k][0] = Perm<n>(0, k).cachedComp(Perm<n>(0, 1));
term_[0][k] = term_[k][0].cachedInverse();
} else {
term_[k][0] = Perm<n>(0, k) * Perm<n>(0, 1);
term_[0][k] = term_[k][0].inverse();
}
for (int j = 1; j < k; ++j) {
if constexpr (cached) {
term_[k][j] = Perm<n>(j, k).cachedComp(Perm<n>(0, j));
term_[j][k] = term_[k][j].cachedInverse();
} else {
term_[k][j] = Perm<n>(j, k) * Perm<n>(0, j);
term_[j][k] = term_[k][j].inverse();
}
}
}
count_[0] = 1;
count_[1] = 1; // this is where A_n differs from S_n
for (int i = 2; i < n; ++i)
count_[i] = i + 1;
// All usable_[k] should be the identity for k != 1.
usable_[1] = Perm<n>(0, 1);
break;
default:
// Each term_[k][j] should be the identity.
std::fill(count_, count_ + n, 1);
for (int i = 1; i < n; ++i)
usable_[i] = Perm<n>(0, i);
break;
}
setup();
}
template PermGroup<2, false>::PermGroup(NamedPermGroup);
template PermGroup<3, false>::PermGroup(NamedPermGroup);
template PermGroup<4, false>::PermGroup(NamedPermGroup);
template PermGroup<5, false>::PermGroup(NamedPermGroup);
template PermGroup<6, false>::PermGroup(NamedPermGroup);
template PermGroup<7, false>::PermGroup(NamedPermGroup);
template PermGroup<8, false>::PermGroup(NamedPermGroup);
template PermGroup<9, false>::PermGroup(NamedPermGroup);
template PermGroup<10, false>::PermGroup(NamedPermGroup);
template PermGroup<11, false>::PermGroup(NamedPermGroup);
template PermGroup<12, false>::PermGroup(NamedPermGroup);
template PermGroup<13, false>::PermGroup(NamedPermGroup);
template PermGroup<14, false>::PermGroup(NamedPermGroup);
template PermGroup<15, false>::PermGroup(NamedPermGroup);
template PermGroup<16, false>::PermGroup(NamedPermGroup);
template PermGroup<2, true>::PermGroup(NamedPermGroup);
template PermGroup<3, true>::PermGroup(NamedPermGroup);
template PermGroup<4, true>::PermGroup(NamedPermGroup);
template PermGroup<5, true>::PermGroup(NamedPermGroup);
template PermGroup<6, true>::PermGroup(NamedPermGroup);
template PermGroup<7, true>::PermGroup(NamedPermGroup);
template PermGroup<8, true>::PermGroup(NamedPermGroup);
template PermGroup<9, true>::PermGroup(NamedPermGroup);
template PermGroup<10, true>::PermGroup(NamedPermGroup);
template PermGroup<11, true>::PermGroup(NamedPermGroup);
template PermGroup<12, true>::PermGroup(NamedPermGroup);
template PermGroup<13, true>::PermGroup(NamedPermGroup);
template PermGroup<14, true>::PermGroup(NamedPermGroup);
template PermGroup<15, true>::PermGroup(NamedPermGroup);
template PermGroup<16, true>::PermGroup(NamedPermGroup);
template <int n, bool cached>
PermGroup<n, cached>::PermGroup(int k) {
// Remember: all permutations not explicitly set here will be
// initialised to the identity.
for (int upper = 1; upper < k; ++upper)
for (int lower = 0; lower < upper; ++lower) {
// These terms are all self-inverse.
term_[upper][lower] = term_[lower][upper] = Perm<n>(lower, upper);
}
for (int i = 0; i < k; ++i)
count_[i] = i + 1;
std::fill(count_ + k, count_ + n, 1);
// Each usable_[0..(k-1)] should be the identity.
for (int i = k; i < n; ++i)
usable_[i] = Perm<n>(0, i);
setup();
}
template PermGroup<2, false>::PermGroup(int);
template PermGroup<3, false>::PermGroup(int);
template PermGroup<4, false>::PermGroup(int);
template PermGroup<5, false>::PermGroup(int);
template PermGroup<6, false>::PermGroup(int);
template PermGroup<7, false>::PermGroup(int);
template PermGroup<8, false>::PermGroup(int);
template PermGroup<9, false>::PermGroup(int);
template PermGroup<10, false>::PermGroup(int);
template PermGroup<11, false>::PermGroup(int);
template PermGroup<12, false>::PermGroup(int);
template PermGroup<13, false>::PermGroup(int);
template PermGroup<14, false>::PermGroup(int);
template PermGroup<15, false>::PermGroup(int);
template PermGroup<16, false>::PermGroup(int);
template PermGroup<2, true>::PermGroup(int);
template PermGroup<3, true>::PermGroup(int);
template PermGroup<4, true>::PermGroup(int);
template PermGroup<5, true>::PermGroup(int);
template PermGroup<6, true>::PermGroup(int);
template PermGroup<7, true>::PermGroup(int);
template PermGroup<8, true>::PermGroup(int);
template PermGroup<9, true>::PermGroup(int);
template PermGroup<10, true>::PermGroup(int);
template PermGroup<11, true>::PermGroup(int);
template PermGroup<12, true>::PermGroup(int);
template PermGroup<13, true>::PermGroup(int);
template PermGroup<14, true>::PermGroup(int);
template PermGroup<15, true>::PermGroup(int);
template PermGroup<16, true>::PermGroup(int);
template <int n, bool cached>
bool PermGroup<n, cached>::contains(Perm<n> p) const {
for (int i = n - 1; i > 0; --i) {
// INV: p fixes all elements > i, and if p is in the group then it has
// a unique representation of the form:
// term_[i][...] * term_[i-1][...] * ... * term_[1][...].
int img = p[i];
if (img == i) {
// We are insisting that term_[i][i] is the identity.
// Nothing more to do other than move down to the next i.
continue;
}
// At this point we must have img < i.
if (term_[i][img].isIdentity()) {
// We cannot map i -> img.
return false;
}
if (cached)
p = term_[img][i].cachedComp(p); /* lhs is inverse */
else
p = term_[img][i] /* inverse term */ * p;
}
// Once we hit i == 0, p must be the identity.
return true;
}
template bool PermGroup<2, false>::contains(Perm<2>) const;
template bool PermGroup<3, false>::contains(Perm<3>) const;
template bool PermGroup<4, false>::contains(Perm<4>) const;
template bool PermGroup<5, false>::contains(Perm<5>) const;
template bool PermGroup<6, false>::contains(Perm<6>) const;
template bool PermGroup<7, false>::contains(Perm<7>) const;
template bool PermGroup<8, false>::contains(Perm<8>) const;
template bool PermGroup<9, false>::contains(Perm<9>) const;
template bool PermGroup<10, false>::contains(Perm<10>) const;
template bool PermGroup<11, false>::contains(Perm<11>) const;
template bool PermGroup<12, false>::contains(Perm<12>) const;
template bool PermGroup<13, false>::contains(Perm<13>) const;
template bool PermGroup<14, false>::contains(Perm<14>) const;
template bool PermGroup<15, false>::contains(Perm<15>) const;
template bool PermGroup<16, false>::contains(Perm<16>) const;
template bool PermGroup<2, true>::contains(Perm<2>) const;
template bool PermGroup<3, true>::contains(Perm<3>) const;
template bool PermGroup<4, true>::contains(Perm<4>) const;
template bool PermGroup<5, true>::contains(Perm<5>) const;
template bool PermGroup<6, true>::contains(Perm<6>) const;
template bool PermGroup<7, true>::contains(Perm<7>) const;
template bool PermGroup<8, true>::contains(Perm<8>) const;
template bool PermGroup<9, true>::contains(Perm<9>) const;
template bool PermGroup<10, true>::contains(Perm<10>) const;
template bool PermGroup<11, true>::contains(Perm<11>) const;
template bool PermGroup<12, true>::contains(Perm<12>) const;
template bool PermGroup<13, true>::contains(Perm<13>) const;
template bool PermGroup<14, true>::contains(Perm<14>) const;
template bool PermGroup<15, true>::contains(Perm<15>) const;
template bool PermGroup<16, true>::contains(Perm<16>) const;
template <int n, bool cached>
bool PermGroup<n, cached>::operator == (const PermGroup& other) const {
// A quick pre-check on count_[], which should be identical.
if (! std::equal(count_, count_ + n, other.count_))
return false;
// Check that every generator of this group belongs to other.
// If so, the groups are equal (since the sizes are the same, so we do
// not need to do the same test in reverse).
for (int k = 1; k < n; ++k) {
// Do not test the last generator term_[k][k], since this is the
// identity and so will pass for free.
for (int i = 0; i < count_[k] - 1; ++i) {
// Examine the following generator:
Perm<n> p = term_[k][usable_[k][i]];
// Our containment test is similar to contains(), but uses
// the fact that we already know that our term fixes k+1,...,n.
// See the contains() implementation for a full explanation.
for (int j = k; j > 0; --j) {
int img = p[j];
if (img == j)
continue;
if (other.term_[j][img].isIdentity())
return false;
if (cached)
p = other.term_[img][j].cachedComp(p); /* lhs is inverse */
else
p = other.term_[img][j] /* inverse term */ * p;
}
}
}
return true;
}
template bool PermGroup<2, false>::operator == (const PermGroup<2, false>&)
const;
template bool PermGroup<3, false>::operator == (const PermGroup<3, false>&)
const;
template bool PermGroup<4, false>::operator == (const PermGroup<4, false>&)
const;
template bool PermGroup<5, false>::operator == (const PermGroup<5, false>&)
const;
template bool PermGroup<6, false>::operator == (const PermGroup<6, false>&)
const;
template bool PermGroup<7, false>::operator == (const PermGroup<7, false>&)
const;
template bool PermGroup<8, false>::operator == (const PermGroup<8, false>&)
const;
template bool PermGroup<9, false>::operator == (const PermGroup<9, false>&)
const;
template bool PermGroup<10, false>::operator == (const PermGroup<10, false>&)
const;
template bool PermGroup<11, false>::operator == (const PermGroup<11, false>&)
const;
template bool PermGroup<12, false>::operator == (const PermGroup<12, false>&)
const;
template bool PermGroup<13, false>::operator == (const PermGroup<13, false>&)
const;
template bool PermGroup<14, false>::operator == (const PermGroup<14, false>&)
const;
template bool PermGroup<15, false>::operator == (const PermGroup<15, false>&)
const;
template bool PermGroup<16, false>::operator == (const PermGroup<16, false>&)
const;
template bool PermGroup<2, true>::operator == (const PermGroup<2, true>&) const;
template bool PermGroup<3, true>::operator == (const PermGroup<3, true>&) const;
template bool PermGroup<4, true>::operator == (const PermGroup<4, true>&) const;
template bool PermGroup<5, true>::operator == (const PermGroup<5, true>&) const;
template bool PermGroup<6, true>::operator == (const PermGroup<6, true>&) const;
template bool PermGroup<7, true>::operator == (const PermGroup<7, true>&) const;
template bool PermGroup<8, true>::operator == (const PermGroup<8, true>&) const;
template bool PermGroup<9, true>::operator == (const PermGroup<9, true>&) const;
template bool PermGroup<10, true>::operator == (const PermGroup<10, true>&)
const;
template bool PermGroup<11, true>::operator == (const PermGroup<11, true>&)
const;
template bool PermGroup<12, true>::operator == (const PermGroup<12, true>&)
const;
template bool PermGroup<13, true>::operator == (const PermGroup<13, true>&)
const;
template bool PermGroup<14, true>::operator == (const PermGroup<14, true>&)
const;
template bool PermGroup<15, true>::operator == (const PermGroup<15, true>&)
const;
template bool PermGroup<16, true>::operator == (const PermGroup<16, true>&)
const;
template <int n, bool cached>
void PermGroup<n, cached>::writeTextLong(std::ostream& out) const {
// We repeat the code for writeTextShort() because we would like to
// hang on to the computed group size for a bit longer.
auto s = size();
out << (s == 1 ? "Trivial" : s == Perm<n>::nPerms ? "Symmetric" :
(s << 1) == Perm<n>::nPerms ? "Alternating" : "Permutation");
out << " group of degree " << n << ", order " << s;
out << std::endl;
if (s == 1)
out << "No generators" << std::endl;
else {
out << "Generators:" << std::endl;
for (int k = 1; k < n; ++k)
if (count_[k] > 1) {
for (int i = 0; i < count_[k] - 1; ++i) {
if (i > 0)
out << ' ';
out << term_[k][usable_[k][i]];
}
out << std::endl;
}
}
}
template void PermGroup<2, false>::writeTextLong(std::ostream&) const;
template void PermGroup<3, false>::writeTextLong(std::ostream&) const;
template void PermGroup<4, false>::writeTextLong(std::ostream&) const;
template void PermGroup<5, false>::writeTextLong(std::ostream&) const;
template void PermGroup<6, false>::writeTextLong(std::ostream&) const;
template void PermGroup<7, false>::writeTextLong(std::ostream&) const;
template void PermGroup<8, false>::writeTextLong(std::ostream&) const;
template void PermGroup<9, false>::writeTextLong(std::ostream&) const;
template void PermGroup<10, false>::writeTextLong(std::ostream&) const;
template void PermGroup<11, false>::writeTextLong(std::ostream&) const;
template void PermGroup<12, false>::writeTextLong(std::ostream&) const;
template void PermGroup<13, false>::writeTextLong(std::ostream&) const;
template void PermGroup<14, false>::writeTextLong(std::ostream&) const;
template void PermGroup<15, false>::writeTextLong(std::ostream&) const;
template void PermGroup<16, false>::writeTextLong(std::ostream&) const;
template void PermGroup<2, true>::writeTextLong(std::ostream&) const;
template void PermGroup<3, true>::writeTextLong(std::ostream&) const;
template void PermGroup<4, true>::writeTextLong(std::ostream&) const;
template void PermGroup<5, true>::writeTextLong(std::ostream&) const;
template void PermGroup<6, true>::writeTextLong(std::ostream&) const;
template void PermGroup<7, true>::writeTextLong(std::ostream&) const;
template void PermGroup<8, true>::writeTextLong(std::ostream&) const;
template void PermGroup<9, true>::writeTextLong(std::ostream&) const;
template void PermGroup<10, true>::writeTextLong(std::ostream&) const;
template void PermGroup<11, true>::writeTextLong(std::ostream&) const;
template void PermGroup<12, true>::writeTextLong(std::ostream&) const;
template void PermGroup<13, true>::writeTextLong(std::ostream&) const;
template void PermGroup<14, true>::writeTextLong(std::ostream&) const;
template void PermGroup<15, true>::writeTextLong(std::ostream&) const;
template void PermGroup<16, true>::writeTextLong(std::ostream&) const;
template <int n, bool cached>
PermGroup<n, cached> PermGroup<n, cached>::centraliser(
const PermClass<n>& conj) {
// Begin with the trivial group.
PermGroup ans;
// The only k with non-trivial term_[k][j] (k > j) are those k that
// occur at the end of a cycle.
// Work through each group of cycles of the same size.
int eltStart = 0; // first element in the first cycle of this group
int eltEnd = 0; // first element in the first cycle of the *next* group
int cycleStart = 0; // first cycle of this group
int cycleEnd = 1; // first cycle of the *next* group
while (true) {
// INV: eltEnd == eltStart
// INV: cycleEnd == cycleStart + 1
int cycleLen = conj.cycle(cycleStart);
eltEnd += cycleLen;
int groupSize = 1;
while (true) {
// Process k as the last element of cycle #(cycleEnd - 1).
// We can move k to any j in the range eltStart ≤ j < k.
int k = eltEnd - 1;
if (k != eltStart) {
ans.count_[k] = eltEnd - eltStart;
ans.usable_[k] = Perm<n>::rot(eltStart);
if (cycleLen == 1) {
// This is a group of fixed points (length 1 cycles),
// so our terms can all just be pair swaps.
for (int j = eltStart; j < k; ++j)
ans.term_[k][j] = ans.term_[j][k] = Perm<n>(j, k);
} else {
int kCycleStart = k + 1 - cycleLen;
std::array<int, n> img;
// Permutations moving k to an earlier cycle in this group:
int j = eltStart;
for (int c = 0; c < groupSize - 1; ++c) {
// Build the cycle ( j j+1 ... j+cycleLen-1 )
for (int i = 0; i < j; ++i)
img[i] = i;
for (int i = j; i < j + cycleLen - 1; ++i)
img[i] = i + 1;
img[j + cycleLen - 1] = j;
for (int i = j + cycleLen; i < n; ++i)
img[i] = i;
Perm<n> shift(img);
// Build a swap between cycles:
// ( j j+1 ... ) <-> ( k kCycleStart ... k-1 )
for (int i = 0; i < j; ++i)
img[i] = i;
img[j] = k;
for (int i = j + 1; i < j + cycleLen; ++i)
img[i] = i + k - j - cycleLen;
for (int i = j + cycleLen; i < kCycleStart; ++i)
img[i] = i;
for (int i = kCycleStart; i < k; ++i)
img[i] = i + j + cycleLen - k;
img[k] = j;
for (int i = k + 1; i < n; ++i)
img[i] = i;
Perm<n> term(img);
for (int i = 0; i < cycleLen; ++i) {
ans.term_[k][j] = term;
ans.term_[j][k] = term.inverse();
term = shift * term;
++j;
}
}
// Permutations moving k within its own cycle:
// Build the cycle ( kCycleStart kCycleStart+1 ... k )
for (int i = 0; i < kCycleStart; ++i)
img[i] = i;
for (int i = kCycleStart; i < k; ++i)
img[i] = i + 1;
img[k] = kCycleStart;
for (int i = k + 1; i < n; ++i)
img[i] = i;
Perm<n> shift(img);
Perm<n> term = shift;
for (int i = 0; i < cycleLen - 1; ++i) {
ans.term_[k][j] = term;
ans.term_[j][k] = term.inverse();
term = shift * term;
++j;
}
// At this point we should have j == k, and we are done.
}
}
if (eltEnd == n)
goto done;
if (conj.cycle(cycleStart) != conj.cycle(cycleEnd))
break;
eltEnd += cycleLen;
++cycleEnd;
++groupSize;
}
// Move on to the next cycle group.
eltStart = eltEnd;
cycleStart = cycleEnd;
++cycleEnd;
}
done:
ans.setup();
return ans;
}
template PermGroup<2, false> PermGroup<2, false>::centraliser(
const PermClass<2>&);
template PermGroup<3, false> PermGroup<3, false>::centraliser(
const PermClass<3>&);
template PermGroup<4, false> PermGroup<4, false>::centraliser(
const PermClass<4>&);
template PermGroup<5, false> PermGroup<5, false>::centraliser(
const PermClass<5>&);
template PermGroup<6, false> PermGroup<6, false>::centraliser(
const PermClass<6>&);
template PermGroup<7, false> PermGroup<7, false>::centraliser(
const PermClass<7>&);
template PermGroup<8, false> PermGroup<8, false>::centraliser(
const PermClass<8>&);
template PermGroup<9, false> PermGroup<9, false>::centraliser(
const PermClass<9>&);
template PermGroup<10, false> PermGroup<10, false>::centraliser(
const PermClass<10>&);
template PermGroup<11, false> PermGroup<11, false>::centraliser(
const PermClass<11>&);
template PermGroup<12, false> PermGroup<12, false>::centraliser(
const PermClass<12>&);
template PermGroup<13, false> PermGroup<13, false>::centraliser(
const PermClass<13>&);
template PermGroup<14, false> PermGroup<14, false>::centraliser(
const PermClass<14>&);
template PermGroup<15, false> PermGroup<15, false>::centraliser(
const PermClass<15>&);
template PermGroup<16, false> PermGroup<16, false>::centraliser(
const PermClass<16>&);
template PermGroup<2, true> PermGroup<2, true>::centraliser(
const PermClass<2>&);
template PermGroup<3, true> PermGroup<3, true>::centraliser(
const PermClass<3>&);
template PermGroup<4, true> PermGroup<4, true>::centraliser(
const PermClass<4>&);
template PermGroup<5, true> PermGroup<5, true>::centraliser(
const PermClass<5>&);
template PermGroup<6, true> PermGroup<6, true>::centraliser(
const PermClass<6>&);
template PermGroup<7, true> PermGroup<7, true>::centraliser(
const PermClass<7>&);
template PermGroup<8, true> PermGroup<8, true>::centraliser(
const PermClass<8>&);
template PermGroup<9, true> PermGroup<9, true>::centraliser(
const PermClass<9>&);
template PermGroup<10, true> PermGroup<10, true>::centraliser(
const PermClass<10>&);
template PermGroup<11, true> PermGroup<11, true>::centraliser(
const PermClass<11>&);
template PermGroup<12, true> PermGroup<12, true>::centraliser(
const PermClass<12>&);
template PermGroup<13, true> PermGroup<13, true>::centraliser(
const PermClass<13>&);
template PermGroup<14, true> PermGroup<14, true>::centraliser(
const PermClass<14>&);
template PermGroup<15, true> PermGroup<15, true>::centraliser(
const PermClass<15>&);
template PermGroup<16, true> PermGroup<16, true>::centraliser(
const PermClass<16>&);
} // namespace regina
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