diff --git a/src/sage/combinat/matrices/latin.py b/src/sage/combinat/matrices/latin.py
index 1a369bba355..53790970c0b 100644
--- a/src/sage/combinat/matrices/latin.py
+++ b/src/sage/combinat/matrices/latin.py
@@ -2496,7 +2496,7 @@ def check_bitrade_generators(a, b, c):
         sage: a, b, c, G = p3_group_bitrade_generators(3)
         sage: check_bitrade_generators(a, b, c)
         True
-        sage: check_bitrade_generators(a, b, gap('()'))
+        sage: check_bitrade_generators(a, b, libgap(gap('()')))
         False
     """
     A = PermutationGroup([a])
diff --git a/src/sage/combinat/permutation.py b/src/sage/combinat/permutation.py
index 3ed9a0a43fd..f1885ad164b 100644
--- a/src/sage/combinat/permutation.py
+++ b/src/sage/combinat/permutation.py
@@ -708,7 +708,7 @@ def _gap_(self, gap):
             sage: gap(Permutation((1,2,3)))                                             # needs sage.libs.gap
             (1,2,3)
             sage: type(_)                                                               # needs sage.libs.gap
-            <class 'sage.interfaces.gap.GapElement'>
+            <class 'sage.libs.gap.element.GapElement_Permutation'>
         """
         return self.to_permutation_group_element()._gap_(gap)
 
diff --git a/src/sage/groups/class_function.py b/src/sage/groups/class_function.py
index c788a596809..e390cbbf288 100644
--- a/src/sage/groups/class_function.py
+++ b/src/sage/groups/class_function.py
@@ -17,12 +17,12 @@
 - Volker Braun (October 2010): Bugfixes, exterior and symmetric power.
 """
 
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2008 Franco Saliola <saliola@gmail.com>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #                  https://www.gnu.org/licenses/
-#*****************************************************************************
+# ****************************************************************************
 
 from sage.structure.sage_object import SageObject
 from sage.structure.richcmp import richcmp, richcmp_method
@@ -66,24 +66,26 @@ class function on the conjugacy classes, in that order.
         sage: chi = ClassFunction(G, values); chi
         Character of Cyclic group of order 4 as a permutation group
     """
+    from sage.misc.superseded import deprecation
     try:
         return group.class_function(values)
     except AttributeError:
         pass
 
-    if isinstance(values, LibGapElement):
+    if isinstance(values, (LibGapElement, tuple, list)):
         return ClassFunction_libgap(group, values)
 
+    # deprecation(36889, "please use libgap class functions instead")
     return ClassFunction_gap(group, values)
 
 
 #####################################################################
-###
-### GAP Interface-based Class Function
-###
-### This is old code that should be deleted once we have transitioned
-### everything to using the library interface to GAP.
-###
+#
+#   GAP Interface-based Class Function
+#
+#   This is old code that should be deleted once we have transitioned
+#   everything to using the library interface to GAP.
+#
 #####################################################################
 
 @richcmp_method
@@ -130,13 +132,13 @@ def __init__(self, G, values):
 
     def _gap_init_(self):
         r"""
-        Returns a string showing how to declare / initialize self in Gap.
+        Return a string showing how to declare / initialize self in Gap.
         Stored in the \code{self._gap_string} attribute.
 
         EXAMPLES::
 
             sage: G = CyclicPermutationGroup(4)
-            sage: values  = [1, -1, 1, -1]
+            sage: values  = gap([1, -1, 1, -1])
             sage: ClassFunction(G, values)._gap_init_()
             'ClassFunction( CharacterTable( Group( [ (1,2,3,4) ] ) ), [ 1, -1, 1, -1 ] )'
         """
@@ -149,7 +151,7 @@ def _gap_(self, *args):
         EXAMPLES::
 
             sage: G = CyclicPermutationGroup(4)
-            sage: values  = [1, -1, 1, -1]
+            sage: values  = gap([1, -1, 1, -1])
             sage: chi = ClassFunction(G, values);  chi
             Character of Cyclic group of order 4 as a permutation group
             sage: type(_)
@@ -173,7 +175,7 @@ def __repr__(self):
 
             sage: G = SymmetricGroup(4)
             sage: values  = [1, -1, 1, 1, -1]
-            sage: ClassFunction(G, values)
+            sage: ClassFunction(G, gap(values))
             Character of Symmetric group of order 4! as a permutation group
         """
         return "Character of %s" % repr(self._group)
@@ -185,7 +187,7 @@ def __iter__(self):
 
         EXAMPLES::
 
-            sage: xi = ClassFunction(SymmetricGroup(4), [1, -1, 1, 1, -1])
+            sage: xi = ClassFunction(SymmetricGroup(4), gap([1, -1, 1, 1, -1]))
             sage: list(xi)
             [1, -1, 1, 1, -1]
         """
@@ -239,7 +241,7 @@ def __reduce__(self):
         EXAMPLES::
 
             sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
-            sage: chi = ClassFunction(G, [1, 1, 1, 1, 1, 1, 1])
+            sage: chi = ClassFunction(G, gap([1, 1, 1, 1, 1, 1, 1]))
             sage: type(chi)
             <class 'sage.groups.class_function.ClassFunction_gap'>
             sage: loads(dumps(chi)) == chi
@@ -249,7 +251,7 @@ def __reduce__(self):
 
     def domain(self):
         r"""
-        Returns the domain of the self.
+        Return the domain of the ``self``.
 
         OUTPUT:
 
@@ -294,7 +296,7 @@ def __call__(self, g):
 
     def __add__(self, other):
         r"""
-        Returns the sum of the characters self and other.
+        Return the sum of the characters self and other.
 
         INPUT:
 
@@ -307,7 +309,7 @@ def __add__(self, other):
 
         EXAMPLES::
 
-            sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
+            sage: chi = ClassFunction(SymmetricGroup(4), gap([3, 1, -1, 0, -1]))
             sage: s = chi+chi
             sage: s
             Character of Symmetric group of order 4! as a permutation group
@@ -321,7 +323,7 @@ def __add__(self, other):
 
     def __sub__(self, other):
         r"""
-        Returns the difference of the characters ``self`` and ``other``.
+        Return the difference of the characters ``self`` and ``other``.
 
         INPUT:
 
@@ -366,7 +368,7 @@ def __mul__(self, other):
         EXAMPLES::
 
             sage: G = SymmetricGroup(4)
-            sage: chi1 = ClassFunction(G, [3, 1, -1, 0, -1])
+            sage: chi1 = ClassFunction(G, gap([3, 1, -1, 0, -1]))
             sage: 3*chi1
             Character of Symmetric group of order 4! as a permutation group
             sage: 3*chi1 == chi1+chi1+chi1
@@ -384,7 +386,7 @@ def __mul__(self, other):
             sage: (zeta3 * chi1).values()
             [3*zeta3, zeta3, -zeta3, 0, -zeta3]
 
-            sage: chi2 = ClassFunction(G, [1, -1, 1, 1, -1])
+            sage: chi2 = ClassFunction(G, gap([1, -1, 1, 1, -1]))
             sage: p = chi1*chi2
             sage: p
             Character of Symmetric group of order 4! as a permutation group
@@ -456,7 +458,7 @@ def __neg__(self):
 
     def __pow__(self, other):
         r"""
-        Returns the product of self with itself other times.
+        Return the product of self with itself other times.
 
         EXAMPLES::
 
@@ -467,13 +469,13 @@ def __pow__(self, other):
             sage: p.values()
             [27, 1, -1, 0, -1]
         """
-        if not isinstance(other, (int,Integer)):
+        if not isinstance(other, (int, Integer)):
             raise NotImplementedError
         return ClassFunction(self._group, self._gap_classfunction ** other)
 
     def symmetric_power(self, n):
         r"""
-        Returns the symmetrized product of self with itself ``n`` times.
+        Return the symmetrized product of self with itself ``n`` times.
 
         INPUT:
 
@@ -486,7 +488,7 @@ def symmetric_power(self, n):
 
         EXAMPLES::
 
-            sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
+            sage: chi = ClassFunction(SymmetricGroup(4), gap([3, 1, -1, 0, -1]))
             sage: p = chi.symmetric_power(3)
             sage: p
             Character of Symmetric group of order 4! as a permutation group
@@ -495,11 +497,11 @@ def symmetric_power(self, n):
         """
         n = Integer(n)
         tbl = gap.UnderlyingCharacterTable(self)
-        return ClassFunction(self._group, gap.SymmetricParts(tbl,[self],n)[1])
+        return ClassFunction(self._group, gap.SymmetricParts(tbl, [self], n)[1])
 
     def exterior_power(self, n):
         r"""
-        Returns the anti-symmetrized product of self with itself ``n`` times.
+        Return the anti-symmetrized product of self with itself ``n`` times.
 
         INPUT:
 
@@ -512,7 +514,7 @@ def exterior_power(self, n):
 
         EXAMPLES::
 
-            sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
+            sage: chi = ClassFunction(SymmetricGroup(4), gap([3, 1, -1, 0, -1]))
             sage: p = chi.exterior_power(3)   # the highest anti-symmetric power for a 3-d character
             sage: p
             Character of Symmetric group of order 4! as a permutation group
@@ -523,11 +525,11 @@ def exterior_power(self, n):
         """
         n = Integer(n)
         tbl = gap.UnderlyingCharacterTable(self)
-        return ClassFunction(self._group, gap.AntiSymmetricParts(tbl,[self],n)[1])
+        return ClassFunction(self._group, gap.AntiSymmetricParts(tbl, [self], n)[1])
 
     def scalar_product(self, other):
         r"""
-        Returns the scalar product of self with other.
+        Return the scalar product of self with other.
 
         EXAMPLES::
 
@@ -544,7 +546,7 @@ def scalar_product(self, other):
 
     def is_irreducible(self):
         r"""
-        Returns True if self cannot be written as the sum of two nonzero
+        Return True if self cannot be written as the sum of two nonzero
         characters of self.
 
         EXAMPLES::
@@ -558,7 +560,7 @@ def is_irreducible(self):
 
     def degree(self):
         r"""
-        Returns the degree of the character self.
+        Return the degree of the character self.
 
         EXAMPLES::
 
@@ -571,7 +573,7 @@ def degree(self):
 
     def irreducible_constituents(self):
         r"""
-        Returns a list of the characters that appear in the decomposition
+        Return a list of the characters that appear in the decomposition
         of chi.
 
         EXAMPLES::
@@ -694,7 +696,7 @@ def tensor_product(self, other):
             sage: chi1.tensor_product(chi3).values()
             [1, -1, 1]
         """
-        return ClassFunction(self._group, gap.Tensored([self],[other])[1])
+        return ClassFunction(self._group, gap.Tensored([self], [other])[1])
 
     def restrict(self, H):
         r"""
@@ -791,9 +793,9 @@ def adams_operation(self, k):
 
 
 #####################################################################
-###
-### Class function using the GAP library
-###
+#
+#   Class function using the GAP library
+#
 #####################################################################
 
 
@@ -834,12 +836,11 @@ def __init__(self, G, values):
         if isinstance(values, LibGapElement) and values.IsClassFunction():
             self._gap_classfunction = values
         else:
-            self._gap_classfunction = libgap.ClassFunction(G._libgap_(),
-                                                           list(values))
+            self._gap_classfunction = libgap.ClassFunction(G, list(values))
         e = self._gap_classfunction.Conductor().sage()
         self._base_ring = CyclotomicField(e)
 
-    def gap(self):
+    def _libgap_(self):
         r"""
         Return the underlying LibGAP element.
 
@@ -850,15 +851,15 @@ def gap(self):
             sage: chi = ClassFunction(G, values);  chi
             Character of Cyclic group of order 4 as a permutation group
             sage: type(chi)
-            <class 'sage.groups.class_function.ClassFunction_gap'>
-            sage: gap(chi)
-            ClassFunction( CharacterTable( Group( [ (1,2,3,4) ] ) ), [ 1, -1, 1, -1 ] )
+            <class 'sage.groups.class_function.ClassFunction_libgap'>
+            sage: libgap(chi)
+            ClassFunction( CharacterTable( Group([ (1,2,3,4) ]) ), [ 1, -1, 1, -1 ] )
             sage: type(_)
-            <class 'sage.interfaces.gap.GapElement'>
+            <class 'sage.libs.gap.element.GapElement_List'>
         """
         return self._gap_classfunction
 
-    _libgap_ = _gap_ = gap
+    gap = _gap_ = _libgap_
 
     def _repr_(self):
         r"""
@@ -877,6 +878,16 @@ def _repr_(self):
         """
         return "Character of %s" % repr(self._group)
 
+    def __hash__(self):
+        r"""
+        TESTS::
+
+            sage: G = SymmetricGroup(5)
+            sage: chi1 = ClassFunction(G,[1,1,1,1,1,1,1])
+            sage: d = {chi1:'trivial'}
+        """
+        return hash((self._group, tuple(self)))
+
     def __iter__(self):
         r"""
         Iterate through the values.
@@ -1162,7 +1173,7 @@ def __pow__(self, other):
             sage: p.values()
             [27, 1, -1, 0, -1]
         """
-        if not isinstance(other, (int,Integer)):
+        if not isinstance(other, (int, Integer)):
             raise NotImplementedError
         return ClassFunction(self._group, self._gap_classfunction ** other)
 
@@ -1189,8 +1200,8 @@ def symmetric_power(self, n):
             [10, 2, -2, 1, 0]
         """
         n = Integer(n)
-        tbl = self._gap_classfunction.UnderlyingCharacterTable(self)
-        return ClassFunction(self._group, tbl.SymmetricParts([self],n)[1])
+        tbl = self._gap_classfunction.UnderlyingCharacterTable()
+        return ClassFunction(self._group, tbl.SymmetricParts([self], n)[0])
 
     def exterior_power(self, n):
         r"""
@@ -1217,8 +1228,8 @@ def exterior_power(self, n):
             True
         """
         n = Integer(n)
-        tbl = self._gap_classfunction.UnderlyingCharacterTable(self)
-        return ClassFunction(self._group, tbl.AntiSymmetricParts([self],n)[1])
+        tbl = self._gap_classfunction.UnderlyingCharacterTable()
+        return ClassFunction(self._group, tbl.AntiSymmetricParts([self], n)[0])
 
     def scalar_product(self, other):
         r"""
diff --git a/src/sage/groups/perm_gps/permgroup_element.pyx b/src/sage/groups/perm_gps/permgroup_element.pyx
index a459451bd6d..493ff9bb8ea 100644
--- a/src/sage/groups/perm_gps/permgroup_element.pyx
+++ b/src/sage/groups/perm_gps/permgroup_element.pyx
@@ -410,7 +410,7 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             sage: k._gap_()
             (1,2)(3,5,6)
             sage: k._gap_().parent()
-            Gap
+            C library interface to GAP
 
         List notation::
 
@@ -835,13 +835,11 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             sage: g._gap_()
             (1,2,3)(4,5)
         """
-        if gap is None:
-            from sage.interfaces.gap import gap
-        return gap(self._gap_init_())
+        return self._libgap_()
 
     def _libgap_(self):
         r"""
-        Returns self as a libgap element
+        Return ``self`` as a libgap element.
 
         EXAMPLES::
 
@@ -867,7 +865,7 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             sage: p_libgap == p_pexpect
             True
             sage: type(p_libgap) == type(p_pexpect)
-            False
+            True
 
         If the permutation element is built from a libgap element, it is cached
         and returned by this function::
diff --git a/src/sage/groups/perm_gps/permgroup_morphism.py b/src/sage/groups/perm_gps/permgroup_morphism.py
index 0a86068790b..c453934141a 100644
--- a/src/sage/groups/perm_gps/permgroup_morphism.py
+++ b/src/sage/groups/perm_gps/permgroup_morphism.py
@@ -84,7 +84,7 @@ def kernel(self):
             sage: G.is_isomorphic(pr1.kernel())
             True
         """
-        return self.domain().subgroup(gap_group=self._gap_().Kernel())
+        return self.domain().subgroup(gap_group=self._libgap_().Kernel())
 
     def image(self, J):
         r"""
@@ -133,10 +133,10 @@ def image(self, J):
         H = self.codomain()
         if J in self.domain():
             J = PermutationGroup([J])
-            G = self._gap_().Image(J)
+            G = self._libgap_().Image(J)
             return H.subgroup(gap_group=G).gens()[0]
         else:
-            G = self._gap_().Image(J)
+            G = self._libgap_().Image(J)
             return H.subgroup(gap_group=G)
 
     def __call__(self, g):
@@ -210,7 +210,7 @@ def _repr_defn(self):
         """
         return str(self._gap_hom).replace('\n', '')
 
-    def _gap_(self, gap=None):
+    def _libgap_(self):
         r"""
         Return a GAP version of this morphism.
 
@@ -220,7 +220,7 @@ def _gap_(self, gap=None):
             sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
             sage: H = G.subgroup([G([(1,2,3,4)])])
             sage: phi = PermutationGroupMorphism_from_gap(H, G, gap.Identity)
-            sage: phi._gap_()
+            sage: phi._libgap_()
             Identity
         """
         return self._gap_hom
@@ -239,7 +239,7 @@ def __call__(self, g):
             sage: [pr1(g) for g in G.gens()]
             [(3,7,5)(4,8,6), (1,2,6)(3,4,8)]
         """
-        return self.codomain()(self._gap_().Image(g))
+        return self.codomain()(self._libgap_().Image(g))
 
 
 class PermutationGroupMorphism_im_gens(PermutationGroupMorphism):
@@ -295,7 +295,7 @@ def _repr_defn(self):
         """
         return "%s -> %s" % (list(self.domain().gens()), self._images)
 
-    def _gap_(self):
+    def _libgap_(self):
         r"""
         Return a GAP representation of this morphism.
 
@@ -304,11 +304,10 @@ def _gap_(self):
             sage: G = CyclicPermutationGroup(4)
             sage: H = DihedralGroup(4)
             sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
-            sage: phi._gap_()
-            GroupHomomorphismByImages( Group( [ (1,2,3,4) ] ), Group(
-            [ (1,2,3,4), (1,4)(2,3) ] ), [ (1,2,3,4) ], [ (1,2,3,4) ] )
+            sage: phi._libgap_()
+            [ (1,2,3,4) ] -> [ (1,2,3,4) ]
         """
-        return self.domain()._gap_().GroupHomomorphismByImages(self.codomain(), self.domain().gens(), self._images)
+        return self.domain()._libgap_().GroupHomomorphismByImages(self.codomain(), self.domain().gens(), self._images)
 
 
 def is_PermutationGroupMorphism(f) -> bool:
diff --git a/src/sage/libs/gap/element.pyx b/src/sage/libs/gap/element.pyx
index e522bf5bc03..58bbe8922d9 100644
--- a/src/sage/libs/gap/element.pyx
+++ b/src/sage/libs/gap/element.pyx
@@ -380,7 +380,7 @@ cdef GapElement make_GapElement(parent, Obj obj) noexcept:
         sage: libgap(None)
         Traceback (most recent call last):
         ...
-        AttributeError: 'NoneType' object has no attribute '_libgap_init_'...
+        AttributeError: 'NoneType' object has no attribute '_gap_init_'...
     """
     cdef GapElement r = GapElement.__new__(GapElement)
     r._initialize(parent, obj)
diff --git a/src/sage/libs/gap/libgap.pyx b/src/sage/libs/gap/libgap.pyx
index 63400adab4c..17a7aca9003 100644
--- a/src/sage/libs/gap/libgap.pyx
+++ b/src/sage/libs/gap/libgap.pyx
@@ -319,7 +319,7 @@ class Gap(Parent):
                 return x._libgap_()
             except AttributeError:
                 pass
-            x = str(x._libgap_init_())
+            x = str(x._gap_init_())
             return make_any_gap_element(self, gap_eval(x))
 
     def _construct_matrix(self, M):
@@ -400,7 +400,7 @@ class Gap(Parent):
         cdef GapElement elem
 
         if not isinstance(gap_command, str):
-            gap_command = str(gap_command._libgap_init_())
+            gap_command = str(gap_command._gap_init_())
 
         initialize()
         elem = make_any_gap_element(self, gap_eval(gap_command))
diff --git a/src/sage/modules/with_basis/invariant.py b/src/sage/modules/with_basis/invariant.py
index 3408e8abf0b..edcd603185f 100644
--- a/src/sage/modules/with_basis/invariant.py
+++ b/src/sage/modules/with_basis/invariant.py
@@ -22,6 +22,7 @@
 from sage.categories.finite_dimensional_modules_with_basis import FiniteDimensionalModulesWithBasis
 from sage.sets.family import Family
 from sage.matrix.constructor import Matrix
+from sage.libs.gap.libgap import libgap
 
 
 class FiniteDimensionalInvariantModule(SubmoduleWithBasis):
@@ -801,25 +802,22 @@ def __classcall_private__(cls, M, G, chi,
 
         Check the :class:`ValueError`::
 
-            sage: from sage.groups.class_function import ClassFunction_libgap
-            sage: chi = ClassFunction_libgap(G, chi)
-            sage: T = M.twisted_invariant_module(G, chi, action_on_basis=action)
+            sage: T = M.twisted_invariant_module(G, "ichigo", action_on_basis=action)
             Traceback (most recent call last):
             ...
             ValueError: chi must be a list/tuple or a class function of the group G
         """
+        from sage.groups.class_function import ClassFunction, ClassFunction_libgap
 
-        from sage.groups.class_function import ClassFunction, ClassFunction_gap
-
-        if isinstance(chi,(list,tuple)):
-            chi = ClassFunction(G, chi)
-        elif not isinstance(chi, ClassFunction_gap):
+        if isinstance(chi, (list, tuple)):
+            chi = ClassFunction(G, libgap(chi))
+        elif not isinstance(chi, ClassFunction_libgap):
             raise ValueError("chi must be a list/tuple or a class function of the group G")
 
         try:
-            is_trivial = all(chi(conj.an_element()) == 1 for conj in G.conjugacy_classes())
-        except AttributeError: # to handle ReflectionGroups
-            is_trivial = all(chi(G(list(conj)[0])) == 1 for conj in G.conjugacy_classes().values())
+            is_trivial = all(chi(next(iter(conj))) == 1 for conj in G.conjugacy_classes())
+        except AttributeError:  # to handle ReflectionGroups
+            is_trivial = all(chi(G(next(iter(conj)))) == 1 for conj in G.conjugacy_classes())
 
         if is_trivial:
             action_on_basis = kwargs.pop('action_on_basis', None)
diff --git a/src/sage/structure/sage_object.pyx b/src/sage/structure/sage_object.pyx
index 10000aef394..39a5487226f 100644
--- a/src/sage/structure/sage_object.pyx
+++ b/src/sage/structure/sage_object.pyx
@@ -755,14 +755,7 @@ cdef class SageObject:
 
     def _libgap_(self):
         from sage.libs.gap.libgap import libgap
-        return libgap.eval(self._libgap_init_())
-
-    def _libgap_init_(self):
-        """
-        For consistency's sake we provide a ``_libgap_init_`` but in most cases
-        we can use the same as ``_gap_init_`` here.
-        """
-        return self._gap_init_()
+        return libgap.eval(self._gap_init_())
 
     def _gp_(self, G=None):
         if G is None:
diff --git a/src/sage/tests/books/judson-abstract-algebra/homomorph-sage.py b/src/sage/tests/books/judson-abstract-algebra/homomorph-sage.py
index 29b25d1efde..abff8d3189c 100644
--- a/src/sage/tests/books/judson-abstract-algebra/homomorph-sage.py
+++ b/src/sage/tests/books/judson-abstract-algebra/homomorph-sage.py
@@ -155,8 +155,8 @@
     sage: tau.kernel()
     Traceback (most recent call last):
     ...
-    RuntimeError: Gap produced error output
-    ...
+    sage.libs.gap.util.GAPError: Error, no method found! Error, no 1st choice method found for `Kernel' on 1 arguments
+    The 1st argument is 'fail' which might point to an earlier problem
 
 ~~~~~~~~~~~~~~~~~~~~~~ ::