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Edit Here is a more concrete question:

Is for all $g,h,x \in G$:

$$J(g,h) = J(gx,hx) = J(xg,xh)$$ ?

from sage.matrix.operation_table import OperationTable
 
def Jaccard(A,B):
    XA = set([ (x,A[x]) for x in range(len(A))])
    XB = set([ (x,B[x]) for x in range(len(A))])
    print(XA,XB)
    return QQ(len(XA.intersection(XB)))/QQ(len(XA.union(XB)))


def distJ(A,B):
    return sqrt(Jaccard(A,A)+Jaccard(B,B)-2*Jaccard(A,B))

import itertools

def Jaccard(A,B):
    pairs = list(itertools.combinations(range(0, len(A)), 2))
    XA = set([ (A[x],A[y]) for x,y in pairs])
    XB = set([ (B[x],B[y]) for x,y in pairs])
    return QQ(len(XA.intersection(XB)))/QQ(len(XA.union(XB)))

def GramMatrix(finiteGroup):
    G = finiteGroup
    O = OperationTable(G,operator.mul,names="elements")
    M = matrix([[ Jaccard(Permutation([xx+1 for xx in x]),Permutation([yy+1 for yy in y])) for x in O.table()] for y in O.table()],ring=QQ) #RDF)
    return M

def regRepr(finiteGroup):
    G = finiteGroup
    O = OperationTable(G,operator.mul,names="elements")
    #print(O.table())
    ll = [ Permutation([xx +1 for xx in x]).to_matrix() for x in O.table()] 
    return ll

rD4 = (regRepr(DihedralGroup(4)))
#print(rD4)
GD4 = GramMatrix(DihedralGroup(4))
vecs = GD4.columns()
print(vecs)
for M in rD4:
    print("--")
    for v in vecs:
      print(v,M*v)
      print(M*v in vecs)

groups = [SymmetricGroup(2),CyclicPermutationGroup(3),CyclicPermutationGroup(4),KleinFourGroup(),CyclicPermutationGroup(5),CyclicPermutationGroup(6),SymmetricGroup(3),QuaternionGroup(),DihedralGroup(5),AlternatingGroup(4),SymmetricGroup(4),DihedralGroup(8)]
for G in groups: #G = DihedralGroup(4)
    print("Group G:=")
    print(G)
    M = GramMatrix(G)
    print("Gram-Matrix:")
    print(M)
    print("cholesky = ")
    print(M.cholesky())
    print("characteristic-polynomial of Gram-Matrix=")
    print(factor(M.charpoly()))
    print("Volume of G:")
    print(sqrt(M.det()))

from sage.matrix.operation_table import OperationTable
import itertools

def Jaccard(A,B):
    pairs = list(itertools.combinations(range(0, len(A)), 2))
    XA = set([ (A[x],A[y]) for x,y in pairs])
    XB = set([ (B[x],B[y]) for x,y in pairs])
    return QQ(len(XA.intersection(XB)))/QQ(len(XA.union(XB)))

def GramMatrix(finiteGroup):
    G = finiteGroup
    O = OperationTable(G,operator.mul,names="elements")
    M = matrix([[ Jaccard(Permutation([xx+1 for xx in x]),Permutation([yy+1 for yy in y])) for x in O.table()] for y in O.table()],ring=QQ) #RDF)
    return M

def regRepr(finiteGroup):
    G = finiteGroup
    O = OperationTable(G,operator.mul,names="elements")
    #print(O.table())
    ll = [ Permutation([xx +1 for xx in x]).to_matrix() for x in O.table()] 
    return ll

rD4 = (regRepr(DihedralGroup(4)))
#print(rD4)
GD4 = GramMatrix(DihedralGroup(4))
vecs = GD4.columns()
print(vecs)
for M in rD4:
    print("--")
    for v in vecs:
      print(v,M*v)
      print(M*v in vecs)

groups = [SymmetricGroup(2),CyclicPermutationGroup(3),CyclicPermutationGroup(4),KleinFourGroup(),CyclicPermutationGroup(5),CyclicPermutationGroup(6),SymmetricGroup(3),QuaternionGroup(),DihedralGroup(5),AlternatingGroup(4),SymmetricGroup(4),DihedralGroup(8)]
for G in groups: #G = DihedralGroup(4)
    print("Group G:=")
    print(G)
    M = GramMatrix(G)
    print("Gram-Matrix:")
    print(M)
    print("cholesky = ")
    print(M.cholesky())
    print("characteristic-polynomial of Gram-Matrix=")
    print(factor(M.charpoly()))
    print("Volume of G:")
    print(sqrt(M.det()))