Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 3 (2010) 59–67
Computation of isomorphisms of coherent
configurations
Izumi Miyamoto
Department of Computer Science and Media Engineering
University of Yamanashi, Kofu 400-8511, Japan
Received 30 September 2007, accepted 4 February 2010, published online 23 February 2010
Abstract
A program computing isomorphisms between association schemes was applied to speed
up the computation of normalizers of permutation groups. A transitive permutation group
forms an association scheme while a permutation group which may not be transitive forms
a coherent configuration. In this paper we discuss the extension of the above program to
compute isomorphisms between coherent configurations and show some typical examples
of computing normalizers.
Keywords: Association scheme, permutation group, algebraic computation.
Math. Subj. Class.: 05E30, 20B40
1 Introduction
Computation of isomorphisms between association schemes were considered by the author
in [5, 6] in order to classify the isomorphism classes of association schemes of small size.
The resulting computer program can be applied to speed up the computation of the normal-
izers of permutation groups [12, 13] and improvements to the program can be found in [14].
However a transitive permutation group forms an association scheme while a permutation
group which may be intransitive forms a coherent configuration. In this paper we discuss
the extension of the program written by the author to compute isomorphisms between co-
herent configurations. Particularly we will show how the program computes isomorphisms
between coherent configurations formed by intransitive permutation groups and will show
some typical examples of computations as applications.
To begin with we used the relation matrix of a coherent configuration to compute the
isomorphisms of the coherent configuration as we used it to compute the isomorphisms of
an association schemes [12, 13]. The entries of the relation matrix are the relation numbers
E-mail address: imiyamoto@yamanashi.ac.jp (Izumi Miyamoto)
Copyright c© 2010 DMFA Slovenije
60 Ars Math. Contemp. 3 (2010) 59–67
of a coherent configuration. But a coherent configuration may have many relations even
though it is of small degree. So if a coherent configuration has more than one fibers (cf.
[8]), we set a number to all the relations besides the fibers satisfying a certain condition.
A coherent configuration can be considered as a graph with labeled edges. It is known
that when we compute the automorphisms of such a graph, we convert it to some kind
of bipartite graph and then we can use some program package for graphs. Our program is
written in GAP Programming Language [4] and GAP has a share package named GRAPE
[19] for computations with graphs. We will also show how the package GRAPE computes
the isomorphisms of coherent configurations in the Appendix.
We should mention about the programming package presented in [3] for computations
with coherent configurations. It computes various properties of coherent configurations.
The UNIX port of the programming system is named “COCO” and available at [2]. In
particular, it computes the order of the automorphism group of a coherent configuration.
Recently it has been modified to compute the automorphisms of a coherent configuration
[18]. Readers may also refer to [7, 9, 15, 16] for information about coherent configurations.
2 Notations and Preliminaries
Let Ω = {1, 2, · · · , n} and let Rk, 1 ≤ k ≤ d, be subsets of Ω× Ω. First we would like to
define the system of axioms for the concept of a coherent algebra and its matrix analogue.
Definition 2.1. (Ω, {Rk}k=1,2,··· ,d) is a coherent configuration if it satisfies the following:
CC1. {R1, R2, · · · , Rd} is a partition of Ω× Ω;
CC2. For some r < d, {R1, R2, · · · , Rr} is a partition of the diagonal {(x, x)|x ∈
Ω} of Ω;
CC3. For every k there exists k∗ such that Rk∗ = {(y, x)|(x, y) ∈ Rk};
CC4. There exist constant numbers pi,j,k such that for any (x, z) ∈ Rk the number
of points y ∈ Ω such that (x, y) ∈ Ri and (y, z) ∈ Rj is equal to pi,j,k.
Let Ak, k = 1, 2, · · · , d, be n× n matrices whose rows and columns are indexed by
Ω and whose (x, y) entry Ak(x, y) is defined by
Ak(x, y) =
{
1 if (x, y) ∈ Rk
0 otherwise.
Then using these matrices, the definition of a coherent configuration takes the following
form.
Definition 2.2. {Ak}k=1,2,··· ,d is a coherent configuration on Ω if the following holds:
CC1.
∑d
k=1 Ak = J ( all 1 matrix );
CC2.
∑r
k=1 Ak = I ( identity matrix ) for some r < d;
CC3. For every k there exists k∗ such that tAk = Ak∗ ;
CC4. AiAj =
∑d
k=1 pi,j,kAk.
We call each Ri a relation and Ai an adjacency matrix. We define the relation matrix A of
a coherent configuration (Ω, {Rk}k=1,2,···d) by
A = 1A1 + 2A2 + · · ·+ dAd.
I. Miyamoto: Computation of isomorphisms of coherent configurations 61
Set Ωk = {x ∈ Ω|(x, x) ∈ Rk} for 1 ≤ k ≤ r in CC2. Then {Ωk}k=1,2,··· ,r forms a
partition of Ω. We collect relations Ri such that Ri ⊆ Ωk×Ωk for each Ωk, 1 ≤ k ≤ r and
denote them {Rk, Rk(1), Rk(2), · · · , Rk(dk)}. Then each (Ωk, {Rk} ∪ {Rk(i)}i=1,2,··· ,dk)
is a coherent configuration and we call it a fiber of the configuration (Ω, {Ri}i=1,2,··· ,d).
A coherent configuration is said to be homogeneous if it has only one fiber, that is, r = 1,
or equivalently if A1 = I . In particular we use the terminology “association scheme” for a
homogeneous coherent configuration.
Let g be a permutation on Ω. Then g acts on Ω × Ω naturally by (x, y)g = (xg, yg)
for (x, y) ∈ Ω × Ω. For a subset R ⊆ Ω × Ω, we define Rg = {(xg, yg)|(x, y) ∈ R}.
Let (Ω, {Rk}k=1,2,··· ,d) and (Ω, {R′k}k=1,2,··· ,d′) be coherent configurations. We say that
they are isomorphic if d = d′ and there exists a permutation g on Ω such that for every Rk
there exists R′k′ such that R
′
k′ = R
g
k. Then we can consider two kinds of automorphisms
of a coherent configuration. One is a permutation on Ω which fixes every relation Rk of a
coherent configuration (Ω, {Rk}k=1,2,··· ,d). The other is a permutation which may move
each relation Rk to another relation Rk′ . These kinds of automorphisms are called by
various names ([8, 1, 10]). The former is called a strict automorphism, a combinatorial
automorphism or simply an automorphism. The latter is called a weak automorphism or a
colored automorphism. We note that the third kind of automorphism is considered in [10].
However in the present paper we call the second kind an automorphism for short.
As we mentioned in the introduction, a popular way to construct coherent configura-
tions and association schemes is from permutation groups. Let G be a permutation group
on Ω. As noted in the previous paragraph, G acts naturally on Ω×Ω. Let {R1, R2, · · · , Rd}
be the orbits of G defined by this action. Then this partition of Ω × Ω makes a coherent
configuration. Here we arrange Rk ⊆ {(x, x)|x ∈ Ω} for 1 ≤ k ≤ r. In particular if G is
transitive, then R1 = {(x, x)|x ∈ Ω} and the partition becomes an association scheme.
3 Computation of isomorphisms between coherent configurations and
automorphisms
Let (Ω, {Ri}i=1,2,··· ,d) be a coherent configuration and let A be the relation matrix of the
configuration. We may sometimes abuse notation by calling the relation matrix A a coher-
ent configuration. Our algorithm to compute isomorphisms and automorphisms is based
on that in [13]. We use relation matrices of coherent configurations to compute with them.
However, as is seen in an example in the following section, a coherent configuration may
have too many relations, even if its degree n is not so large. So we modify the relation
matrix using the fibers of the configuration. Let {Ωk}k=1,2,··· ,r be the partition of Ω defin-
ing the fibers of a coherent configuration (Ω, {Ri}i=1,2,··· ,d), where its adjacency matrices
satisfy
∑r
i=1 Ai = I (identity matrix). We consider the union of the relations Ri such that
Ri = Ωj × Ωk for some subsets Ωj and Ωk stated above with j, k ≤ r and j 6= k. We
set R′d′+1 to be the union of such relations, if they exist, where the remaining relations are
R′1, R
′
2, · · · , R′d′ satisfying R′i 6= Ωj × Ωk for any j, k ≤ r, j 6= k. We use the relation
matrix A′ made by the partition {R′1, R′2, · · · , R′d′ , R′d′+1} to compute the automorphisms
of the coherent configuration. We may arrange the relations of A so that Ri = R′i if
1 ≤ i ≤ d′ and Ri = Ωj × Ωk for some j, k ≤ r, j 6= k if d′ + 1 ≤ i ≤ d. The following
62 Ars Math. Contemp. 3 (2010) 59–67
is an example of A and A′.
A =

1 4 7 9 11 11
4 1 9 7 11 11
8 10 2 5 13 13
10 8 5 2 13 13
12 12 14 14 3 6
12 12 14 14 6 3
 A
′ =

1 4 7 9 11 11
4 1 9 7 11 11
8 10 2 5 11 11
10 8 5 2 11 11
11 11 11 11 3 6
11 11 11 11 6 3

Here A is formed by a group G =Group((1, 2)(3, 4), (5, 6)). A has 3 fibers and 14 re-
lations. Ω1 = {1, 2}, Ω2 = {3, 4} and Ω3 = {5, 6}. The relations Ri, i ≥ 7 are not
contained in the fibers. Among these relations, if i ≥ 11, Ri = Ωj×Ωk for (j, k) = (1, 3),
(2, 3), (3, 1) or (3, 2), while R7 ∪R9 = Ω1 × Ω2 and R8 ∪R10 = Ω2 × Ω1. So Ri = R′i
for 1 ≤ i ≤ 10 and R′11 = R11 ∪R12 ∪R13 ∪R14.
Let g be an automorphism of A′ and we will show that g is an automorphism of A. First
we note that R′gd′+1 = R
′
d′+1, since any Ri is contained in some Ωj ×Ωk. For any relation
Ri of A, let (x, y) ∈ Ri and suppose that (xg, yg) ∈ Ri′ for some relation Ri′ of A. Then
we will see Rgi = Ri′ . If i ≤ d′, then Ri = R′i and we have R
g
i = R
′g
i = R
′
i′ = Ri′ .
If i ≥ d′ + 1, there exist subsets Ωj and Ωk stated above for some j, k ≤ r, j 6= k such
that (x, y) ∈ Ri = Ωj × Ωk. Let xg ∈ Ωj′ and let yg ∈ Ωk′ . Then considering the action
of g on the diagonals of the fibers, we see that {(z, z)|z ∈ Ωj}g = {(z′, z′)|z′ ∈ Ωj′}
and {(w, w)|w ∈ Ωk}g = {(w′, w′)|w′ ∈ Ωk′}. This implies that Rgi = (Ωj × Ωk)g =
Ωj′×Ωk′ = Ri′ . Therefore g becomes an automorphism obtained from the relation matrix
A of the configuration (Ω, {Ri}i=1,2,··· ,d). This computation is shown in Example 3 in the
next section.
We apply our program to compute normalizers and the results of our experiments are
shown in the next section. In [12, 13] the automorphism groups of the association schemes
formed by the actions on the orbits of an intransitive group are computed and the direct
product of them together with the isomorphisms among the association schemes is consid-
ered, while this time we can compute the automorphism group of the coherent configura-
tion formed by the intransitive group directly. We note that there exist not a few groups
of which normalizers are hard to compute even though they are of small degree. Gener-
ally our program computes normalizers of such groups smoothly except for a few groups.
However Example 1 in the next section may explain the difficulty of the computation of
normalizers. In contrast to normalizers, automorphism groups of coherent configurations
were computed quickly if they do not have a large number of relations.
The automorphisms of coherent configurations can be computed by a program package
for computations with graphs suggested by [17]. We have done an experiment using the
program package GRAPE. The computation is shown in the Appendix.
4 Experiments
Let G be a permutation group on the set Ω and let A be the coherent configuration formed
by G. Let Sym(n) denote the symmetric group of degree n = |Ω|. The normalizer of
a group G in a group H is defined by NORM(H,G) = {h ∈ H|h−1Gh = G}. The
normalizer permutes the orbits of G on Ω × Ω. So the normalizer of G in Sym(n) is
contained in the automorphism group of the coherent configuration A formed by G by
definition.
I. Miyamoto: Computation of isomorphisms of coherent configurations 63
We would like to explain how the GAP-default function NORMALIZER, which will be
abbreviated as NORM below, computes a normalizer in the symmetric group briefly in an
easy example. Suppose that G has two orbits of different size n1 and n2, n1 + n2 = n.
GAP computes the actions G1 and G2 of G on the two orbits of G. Then it computes
Ni =NORM(Sym(ni), Gi) for i = 1, 2. Moreover if G2 is imprimitive on the orbit of
size n2 with only one block system of block size n′, then the normalizer N2 of G2 in
Sym(n2) is computed in advance by NORM in the group WREATHPRODUCT(Sym(n′)×
Sym(n2/n′)). After these computations it computes the normalizer of G in N1 × N2
by using another GAP-function AUTOMORPHISMGROUPPERMGROUP, which computes
a normalizer directly.
Example 1. Let K =PRIMITIVEGROUP(112, 1) ∼= PSU(4, 3) in the GAP library, and
let G =STABILIZER(K, 112) = {g ∈ K|112g = 112}. Then we have the following data.
G is of degree n = |Ω| = 111 and of order |G| =29160. G has two orbits of length 81 and
30, which are denoted by Ω1 and Ω2 respectively. G is faithful on both of the orbits. We
compute the normalizer N of G in Sym(111) by various methods.
First, we compute it directly by the GAP-function AUTOMORPHISMGROUPPERM-
GROUP. It took less than 0.5 seconds for this computation.
Second, we use the GAP-default function NORM. It took too long for this computa-
tion, so we interrupted the computation and we looked into each step of this computation
explained above. The action G2 of G on Ω2 is imprimitive and has only one block system
of block size 3. So GAP computes N2 =NORM(WREATHPRODUCT(Sym(3), Sym(10)),
G2). But it took about 4 hours for this computation. After this computation it took about
15 seconds to compute the normalizer N =NORM(Sym(81)×N2, G).
Third, we used the program for computation of association schemes [12, 13]. G has
two orbits Ω1 and Ω2 and the actions of G on these orbits are denoted by G1 and G2,
respectively as above. So we have two association schemes A1 and A2 formed by G1
and G2, we can compute their automorphism groups Aut(A1) and Aut(A2) by the pro-
gram in [12, 13] and we will compute Ni =NORM(Aut(Ai), Gi) for i = 1, 2. Then
we can compute N =NORM(N1 × N2, G). Following this line of computation, we ob-
tained Aut(A1) of order 233280 and Aut(A2) =WREATHPRODUCT (Sym(3), Sym(10))
of order 219419659468800. At this step we should compute the same difficult normalizer
N2 =NORM(WREATHPRODUCT(Sym(3), Sym(10)), G2) as in the second computation.
The remaining computation was done quickly.
Finally, we construct a coherent configuration A from G and use our program to com-
pute the automorphism group Aut(A) of A. G has 11 orbits on Ω × Ω. The relation
matrix A is of size 111 × 111 = 12321. The matrix for the constants (pi,j,k) is of size
11×11×11 = 1331. It took 7 seconds to compute Aut(A) by our program and |Aut(A)| =
233280. Then the computation of N =NORM(Aut(A), G) was done quickly. We note that
G is normal in Aut(A) in this example.
Example 2. Let G be the group generated by the permutations
(3, 16)(4, 15)(5, 17)(6, 18)(7, 8)(9, 21)(10, 22)(11, 12)(19, 20)(23, 24),
(1, 14)(2, 13)(5, 6)(7, 8)(9, 22)(10, 21)(11, 24)(12, 23)(17, 18)(19, 20),
(1, 2)(3, 15)(4, 16)(7, 20)(8, 19)(9, 21)(10, 22)(11, 24)(12, 23)(13, 14).
64 Ars Math. Contemp. 3 (2010) 59–67
Then G is elementary abelian of order 8 and degree 24, There exist 6 orbits of G of
length 4. So NORM(Sym(24), G) = NORM(WREATHPRODUCT(Sym(4), Sym(6)), G).
It took about 3 minutes to compute this normalizer. Let A be the coherent configuration
formed by G. Then Aut(A) is of order 98304. It took 3 seconds to compute Aut(A)
and NORM(Aut(A), G). The 6 association schemes which consist of the fibers of A
are mutually isomorphic and their automorphism groups are Sym(4). So if we do not
use a coherent configuration but use association schemes as in [12], we have to compute
NORM(WREATHPRODUCT(Sym(4), Sym(6)), G), which is the same as by the GAP-
function NORM.
Example 3. Let H =TRANSITIVEGROUP(30, 1075) in the GAP library and let G =
{g ∈ H|1g = 1 and 15g = 15}, the stabilizer of the points 1 and 15 in H . G sta-
bilizes two more points. G is of order 64 and of degree 26. All the orbits of G are
of length 2. G has 186 orbits on Ω × Ω. So the relation matrix A of the coherent
configuration formed by G is of size 26 × 26 = 676, while the matrix of the constant
(pi,j,k) is of size 186 × 186 × 186 = 6434856. It took about 9 minutes to compute
Aut(A), and |Aut(A)| = 23781703680. Here we consider the partition {R′i}i=1,··· ,35
introduced in the previous section associated with this case. Let A′ be the relation ma-
trix defined by this partition. Then it took 1 second to compute Aut(A′). In this case it
took about 100 seconds to compute each of NORM(Aut(A), G), NORM(Sym(26), G) and
AUTOMORPHISMGROUPPERMGROUP(Sym(26), G).
As for computing the normalizers of permutation groups, the author considered var-
ious methods [12, 14]. In [12], as is shown in Examples 1 and 2, we construct an as-
sociation scheme from each orbit of an intransitive group, compute the automorphism
groups and consider the direct product of them and the isomorphisms among the asso-
ciation schemes. Here in our program we construct a coherent configuration from the
intransitive group and compute the automorphism group instead of the computations relat-
ing the association schemes stated above. In Table 1, CCN denotes our method to compute
normalizers using an automorphism group of a coherent configuration, ASN denotes the
method to compute normalizers using the direct product of automorphism groups of as-
sociation schemes and the isomorphisms among them shown in [12] and NORM denotes
the GAP function to compute normalizers. Our program CCN took more time in some
cases in Table 1. We explain briefly about these computations. When we compute the
normalizer of H =TRANSITIVEGROUP(27, 1799), we use the normalizer of the subgroup
G = {g ∈ H|1g = 1 and 2g = 2} of H . The orbits of G are {{3, 7}, {4, 8}, {5, 6},
{10, 16, 11, 15, 17, 12, 13, 18, 14}, {19, 24, 27, 20, 23, 25, 26, 22, 21}}. Here we see the
actions of G on the orbits of length 9. The actions of G on the two orbits of length 9
are E(9):2D 8 of order 144, which is doubly transitive, and M(9)=E(9):Q 8 of order 72,
which is also doubly transitive. So the direct sum of two association schemes of degree 9
coincides with the coherent configuration of degree 18 in this case, and the automorphism
group interchanges these two orbits of length 9. But the normalizer leaves these two orbits
fixed, since the two actions are not mutually isomorphic. In ASN, we check the orders of
the actions, while in CCN, we simply compute the automorphism group Aut of the coher-
ent configuration. This is why it took more time for the computation by CCN in this case.
We note that Aut was always easily computed within 0.2 seconds, but it was still hard to
compute NORM(Aut, G) in some cases. These facts explain the times in Table 1.
I. Miyamoto: Computation of isomorphisms of coherent configurations 65
n k CCN ASN NORM n k CCN ASN NORM
24 2827 7 145 192 24 24924 5764 0.5 1852
24 2951 0.5 66 >10h 27 1799 10755 0.5 >10h
24 7232 0.7 145 387 27 1822 11160 0.5 >10h
27 333 0.5 142 348 28 413 813 6 446
27 817 2 125 14040 30 494 1285 0.2 0.2
28 160 1 4004 2694 30 512 1255 0.3 1
28 238 1 2707 1401 30 527 1288 0.4 1
28 273 2 5974 3588 30 528 1277 0.3 0.2
28 346 3 5044 1939 30 1075 827 0.3 0.2
30 143 4 416 >10h 30 1083 807 0.3 0.3
30 628 4 151 304 30 1103 816 0.3 0.2
30 1268 25 433 >10h 30 4898 13742 0.7 >10h
30 1721 5 367 >10h 30 4899 55084 0.7 >10h
Table 1: Computing times of some normalizers of TRANSITIVEGROUP(n, k) in Sym(n).
(In seconds, “>10h” means more than 10 hours.)
5 Concluding remarks
We use an array of size d3 to compute the permutations of the relations {Ri} induced from
isomorphisms. It is hard to compute the array of pi,j,k of size d3, because d can be large
compared to n the size of A. Even though we consider the partition {R′i}, the size d′3 can
still be large. For instance, since most of pi,j,k’s are 0, in GAP, an array is used such that
the (i, j) entry consists of two lists where the indices k with pi,j,k 6= 0 and the values pi,j,k
are stored. So we think that some improvement can be made in computing the permutation
on {R′i}.
6 Appendix
A coherent configuration is a labeled graph and we permit permutation of the labels in
computing its automorphisms. Similarly, as to hypergraphs, the edges and the labels of a
coherent configuration are converted to vertices of some kind of bipartite graph and we use
a program computing automorphism groups of graphs in order to compute automorphism
groups of coherent configurations. NAUTY [11] is a program for computations with graphs,
which computes automorphism groups of graphs and digraphs. GRAPE [19] is a program
package which provides the interface to NAUTY for the GAP system. We have done an ex-
periment to compute automorphism groups of association schemes A formed by transitive
permutation groups G of small degree in the GAP library. The following rough sketch of
our program using GRAPE is suggested by [17].
VERTICES are tuples:
[1], [2], · · · , [n] for original points of Ω,
ARRANGEMENTS([1..n], 2) for distinct ordered pairs of points,
[n + 1], [n + 2], · · · , [n + d] for relations.
EDGES:
[i]→ [i, j], [i, j]→ [j],
[i]→ [n + k] if (i, i) ∈ Rk or equivalently A(i, i) = k,
66 Ars Math. Contemp. 3 (2010) 59–67
[i, j]→ [n + k] if (i, j) ∈ Rk or equivalently A(i, j) = k.
Ãut =AUTGROUPGRAPH(G,VERTICES,ONTUPLES,
FUNCTION(u, v) return ISEDGE(u, v,A);end).
Aut =ACTION(Ãut, [1..n]).
In most cases the automorphism groups Aut can be computed very quickly within 0.2
seconds. But some association schemes are not computed within a reasonable time. For in-
stance, the association scheme formed by WREATHPRODUCT(TRANSITIVEGROUP(6, 4),
TRANSITIVEGROUP(4, 2)) takes more than 4400 minutes to compute its automorphism
group. In the following groups the automorphism groups of the association schemes formed
by these groups could not be computed within 5 minutes and computations were stopped:
• WREATHPRODUCT(Sym(4), TRANSITIVEGROUP(4, 2));
• WREATHPRODUCT(Sym(4), TRANSITIVEGROUP(5, 2));
• WREATHPRODUCT(TRANSITIVEGROUP(6, 7), TRANSITIVEGROUP(4, 2)).
Our program computed these automorphism groups within 0.2 seconds each.
7 Acknowledgment
The author would like to express his thanks to Prof. Ted Dobson for his kind advice.
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