Nonabelian Groups with (96, 20, 4) Difference Sets
Omar A. AbuGhneim
Department of Mathematics
University of Jordan
Amman, Jordan

Ken W. Smith
Department of Mathematics
Central Michigan University
Mt. Pleasant, MI 48859

Submitted: Apr 10, 2006; Accepted: Oct 10, 2006; Published: Jan 3, 2007
AMS Subject Classification: 05B10
Abstract
We resolve the existence problem of (96, 20, 4) difference sets in 211 of 231 groups
of order 96. If G is a group of order 96 with normal subgroups of orders 3 and 4 then
by first computing 32- and 24-factor images of a hypothetical (96, 20, 4) difference
set in G we are able to either construct a difference set or show a difference set does
not exist.
Of the 231 groups of order 96, 90 groups admit (96, 20, 4) difference sets and 121
do not. The ninety groups with difference sets provide many genuinely nonabelian
difference sets. Seven of these groups have exponent 24.
These difference sets provide at least 37 nonisomorphic symmetric (96, 20, 4)
designs.
1 Introduction
A (v, k, λ) difference set is a subset D of size k in a group G of order v with the property
that for every nonidentity g in G, there are exactly λ ordered pairs (x, y) ∈ D × D such
that
xy
−1
= g.

D
ˆ
D
(−1)
= (k − λ)1
G
+ λ
ˆ
G. (1)
If a group G has a difference set D then {gD : g ∈ G} is the set of blocks of a
symmetric (v, k, λ) design with point set G. On this design G acts by left multiplication
as a sharply transitive automorphism group. Conversely, any symmetric design with a
sharply transitive automorphism group on points is isomorphic to a design constructed
from the set of left translates of a difference set. A difference set is said to be “genuinely
nonabelian” if the underlying design has no abelian group acting regularly on points.
Difference sets with parameters (q
d+1
(
q
d+1
−1
q−1
+ 1), q
d
q
d+1
−1
q−1
, q
d

P AQ = I
r
0
⊕ 2I
r
1
⊕ 4I
r
2
⊕ 8I
r
3
⊕ 16I
r
4
⊕ 80I
r
5
.
Since the prime 5 divides k = 20 but does not divide k − λ = 16, then the rank of A over
GF (5) is 95 and r
5
= 1. Thus the sum of the other matrix sizes, r
0
+ r
1
+ r
2
+ r
3

induces, by linearity, a homomorphism from Z[G]
onto Z[G

]. If the kernel of f is the subgroup U , let T be a complete set of distinct
representatives of cosets of U and, for g ∈ T , set t
g
:= |gU ∩D|. The multiset {t
g
: g ∈ T }
is the collection of “intersection numbers” of D with respect to U. The image of
ˆ
D under
the function f is f(
ˆ
D) =

g∈T
t
g
f(g). This group ring element satisfies the equation
f(
ˆ
D)f(
ˆ
D)
(−1)
= (k − λ)1
G

+ λ|U|

i=1
t
i
= 20,
24

i=1
t
i
2
= 32
is a list {t
i
: 1 ≤ i ≤ 24} with a single 4, sixteen 1s, and seven 0s. When this occurs, one
can show that there is a subgroup H

< G

of order eight such that
ˆ
f(D) is equivalent to
(4)1
G

+ (
ˆ
G

−
ˆ

d
q
d+1
−1
q−1
and λ =
q
d
q
d
−1
q−1
) in abelian groups with an elementary abelian subgroup of order q
d+1
. McFarland’s
construction (with q = 4, d = 1) gives (96, 20, 4) difference sets in GAP[96,231]
∼
=
Z
4
2
× Z
6
and GAP[96,220]
∼
=
Z
3
2
× Z

subgroup. These groups are GAP[96,i] where i in the set {70, 159, 160, 162, 167, 194,
195, 196, 197, 218, 219, 220, 221, 226, 227, 228, 229, 230, 231}.
In recent work, [9], Golemac, Mandi´c and Vuˇciˇci´c have constructed (96, 20, 4) difference
sets in 22 nonabelian groups. These groups are GAP[96,i] where i in the set {13, 41, 64,
70, 71, 87, 144, 159, 160, 167, 185, 186, 188, 190, 194, 195, 196, 197, 226, 227, 228, 229}.
A result of Turyn ([17]) rules out the existence of difference sets in GAP[96,2]
∼
=
Z
96
and GAP[96,59]
∼
=
Z
2
× Z
48
.
Arasu, Davis, Jedwab, Ma and McFarland, [3], ruled out the existence of difference sets
in the last two abelian groups, GAP[96,46]
∼
=
Z
4
×Z
24
and GAP[96,176]
∼
=
Z

Grinnell College) used (16, 6, 2) difference sets to construct images (96, 20, 4) difference
sets in groups of order 32 and then used those images to construct (96, 20, 4) difference
sets in GAP[96,221] and GAP[96,231]. The difference sets found during this search had
the same 2-rank as difference sets which had been previously discovered.
the electronic journal of combinatorics 14 (2007), #R8 4
3 New Results on the existence of (96,20,4) Differ-
ence sets
Suppose G is a group of order 96 which has normal subgroups of order 3 and 4 and
suppose that D is a difference set in G. We used GAP to build the 4-images from the
2-images then again the 8-, 16-, 32-images from the 4-, 8-, 16-images respectively. In a
similar way we computed the 24-images. We wrote programs that combined the 32- and
24-images to construct difference sets in G or to show such a difference set does not exist
after exhaustive search. One can find these programs with examples to explain them in
the dissertation of the first author, [2], and on the webpage of the second author, see [16].
We note that these programs give, by exhaustive search, all possible (96, 20, 4) difference
sets in some 72 groups groups of order 96, those groups that have both factor groups of
order 32 and 24.
Table 1 lists the Groups of order 96 which admit a (96, 20, 4) difference sets and have
factor groups of orders 32 and 24. In the first column of Table 1 appears the catalogue
number i of the group [96, i], according to the GAP SmallGroup library of groups of order
96. (The GAP command “e := Elements(SmallGroup(96, i));” can be used to create a
list e of the elements of the group G, indexed from 1 to 96.) The indices of the elements
of the difference set are provided in the second column of Table 1. The third column
provides, in abbreviated form (r
0
, r
1
, r
2
, r

The invariant factors obtained from the abelian difference sets are (30, 1, 34, 1, 29),
(30, 2, 32, 2, 29), (32, 3, 26, 3, 31), (32, 4, 24, 4, 31) and (34, 4, 20, 4, 33). These invariant
factors are from Table 1 and [1]. All difference sets with other invariant factors are
genuinely nonabelian.
Table 1. List of (96, 20, 4) Difference Sets
i Elements of D Invariant factors
10 1, 2, 3, 5, 8, 9, 16, 32, 37, 38, 48, (34, 6, 16, 6, 33)
53, 57, 65, 66, 67, 73, 74, 76, 87
1, 2, 3, 5, 8, 9, 12, 22, 27, 30, 37, 38, (36, 6, 12, 6, 35)
46, 65, 68, 69, 74, 86, 90, 95
13 1, 2, 3, 4, 10, 12, 24, 26, 33, 38, 40, (26, 6, 32, 6, 25)
43, 46, 55, 57, 67, 77, 81, 82, 83
1, 2, 3, 4, 10, 12, 19, 21, 24, 26, 38, 44, (32, 2, 28, 2, 31)
45, 46, 48, 60, 78, 87, 90, 95
1, 2, 3, 4, 10, 12, 14, 26, 33, 50, 55, 57, (26, 10, 24, 10, 25)
68, 69, 71, 73, 81, 82, 83, 92
1, 2, 3, 4, 7, 11, 27, 28, 38, 44, 51, 56, 57, (28, 2, 36, 2, 27)
58, 59, 65, 79, 82, 85, 96
1, 2, 3, 4, 7, 11, 14, 28, 30, 44, 53, 56, 57, (28, 6, 28, 6, 27)
59, 64, 65, 75, 77, 80, 94
1, 2, 3, 4, 7, 10, 22, 24, 26, 27, 38, 43, 51, 57, (30, 6, 24, 6, 29)
58, 69, 70, 71, 73, 88
1, 2, 3, 4, 7, 10, 12, 14, 26, 37, 44, 50, 60, (32, 0, 32, 0, 31)
75, 77, 78, 79, 85, 92, 93
1, 2, 3, 4, 7, 10, 12, 14, 26, 37, 44, 50, 60, (28, 4, 32, 4, 27)
66, 68, 75, 78, 85, 87, 88
1, 2, 3, 4, 7, 10, 12, 14, 26, 37, 44, 45, 47, (30, 2, 32, 2, 29)
50, 60, 72, 73, 75, 78, 85
1, 2, 3, 4, 7, 10, 12, 14, 26, 37, 41, 44, 50, (28, 0, 40, 0, 27)
60, 69, 75, 78, 85, 86, 95

69, 75, 77, 79, 82, 88, 96
1, 2, 3, 4, 5, 16, 18, 25, 37, 58, 62, 63, (34, 6, 16, 6, 33)
64, 80, 81, 86, 88, 90, 91, 95
52 1, 2, 3, 4, 8, 10, 18, 19, 27, 35, 41, 43, (34, 8, 12, 8, 33)
45, 48, 58, 71, 82, 87, 90, 96
1, 2, 3, 4, 8, 10, 17, 18, 19, 27, 41, 45, (36, 8, 8, 8, 35)
48, 64, 65, 74, 83, 89, 91, 94
1, 2, 3, 4, 5, 8, 16, 19, 32, 43, 46, 49, (34, 10, 8, 10, 33)
55, 58, 63, 70, 73, 86, 92, 94
54 1, 2, 3, 4, 6, 12, 24, 29, 38, 39, 47, (36, 6, 12, 6, 35)
50, 55, 58, 60, 66, 80, 82, 83, 94
1, 2, 3, 4, 6, 11, 14, 21, 23, 42, 45, 50, (34, 6, 16, 6, 33)
51, 58, 65, 71, 74, 75, 83, 86
1, 2, 3, 4, 5, 21, 22, 34, 35, 40, 51, (34, 4, 20, 4, 33)
63, 70, 73, 76, 78, 81, 82, 89, 90
1, 2, 3, 4, 5, 8, 16, 32, 37, 38, 46, 50, 55, (34, 3, 22, 3, 33)
65, 66, 68, 82, 83, 93, 95
75 1, 2, 3, 4, 7, 9, 21, 22, 24, 25, 33, 35, (34, 4, 20, 4, 33)
41, 46, 48, 53, 57, 71, 87, 90
1, 2, 3, 4, 7, 8, 13, 21, 22, 41, 46, 47, (32, 8, 16, 8, 31)
56, 72, 74, 80, 84, 92, 94, 95
the electronic journal of combinatorics 14 (2007), #R8 7
77 1, 2, 3, 4, 7, 9, 16, 24, 43, 45, 52, 55, (34, 4, 20, 4, 33)
63, 65, 71, 72, 83, 84, 89, 92
78 1, 2, 3, 4, 7, 13, 21, 22, 23, 26, 27, 46, (34, 3, 22, 3, 33)
49, 56, 58, 60, 68, 72, 75, 95
1, 2, 3, 4, 7, 12, 25, 37, 40, 41, 43, (32, 4, 24, 4, 31)
49, 56, 58, 63, 68, 69, 76, 78, 88
1, 2, 3, 4, 7, 12, 13, 26, 39, 44, 49, (30, 8, 20, 8, 29)
56, 57, 66, 76, 78, 85, 86, 87, 95

87 1, 2, 3, 4, 7, 12, 24, 29, 35, 40, 43, 52, (30, 2, 32, 2, 29)
53, 65, 66, 81, 86, 87, 90, 95
1, 2, 3, 4, 7, 12, 24, 25, 36, 44, 54, 56, (30, 8, 20, 8, 29)
57, 66, 67, 75, 85, 86, 87, 95
the electronic journal of combinatorics 14 (2007), #R8 8
1, 2, 3, 4, 7, 12, 24, 25, 29, 40, 57, 63, (32, 4, 24, 4, 31)
65, 66, 70, 75, 78, 86, 87, 95
1, 2, 3, 4, 7, 12, 24, 25, 29, 40, 41, 57, (32, 2, 28, 2, 31)
63, 65, 68, 69, 70, 75, 78, 88
1, 2, 3, 4, 7, 10, 11, 12, 31, 34, 36, 37, (30, 7, 22, 7, 29)
39, 58, 70, 78, 80, 82, 90, 91
1, 2, 3, 4, 7, 10, 11, 12, 31, 34, 35, 37, (32, 3, 26, 3, 31)
40, 57, 70, 78, 79, 83, 90, 92
88 1, 2, 3, 4, 7, 21, 22, 26, 28, 32, 36, 46, 49, (34, 2, 24, 2, 33)
54, 58, 69, 71, 73, 87, 90
1, 2, 3, 4, 7, 21, 22, 26, 28, 32, 36, 46, (34, 3, 22, 3, 33)
48, 49, 54, 71, 76, 81, 87, 88
1, 2, 3, 4, 7, 21, 22, 26, 28, 32, 36, 45, (34, 4, 20, 4, 33)
46, 49, 54, 69, 73, 76, 81, 95
90 1, 2, 3, 4, 7, 21, 22, 25, 29, 36, 46, 48, (32, 8, 16, 8, 31)
49, 59, 68, 77, 79, 84, 87, 89
1, 2, 3, 4, 7, 19, 25, 35, 40, 41, 56, (32, 4, 24, 4, 31)
61, 68, 69, 74, 77, 80, 83, 85, 88
1, 2, 3, 4, 7, 19, 23, 25, 35, 40, 47, 48, (32, 3, 26, 3, 31)
56, 61, 73, 74, 77, 80, 83, 85
1, 2, 3, 4, 7, 16, 28, 44, 45, 49, 52, 61, (34, 4, 20, 4, 33)
63, 64, 71, 72, 80, 83, 89, 92
1, 2, 3, 4, 7, 10, 11, 12, 19, 31, 40, 51, (32, 2, 28, 2, 31)
53, 58, 61, 63, 64, 80, 94, 96
91 1, 2, 3, 4, 7, 21, 22, 25, 33, 35, 46, 47, (34, 3, 22, 3, 33)

99 1, 2, 3, 4, 7, 13, 19, 25, 41, 50, 55, 56, 62, (34, 4, 20, 4, 33)
67, 68, 69, 82, 85, 88, 96
1, 2, 3, 4, 7, 9, 16, 25, 33, 37, 50, 57, 59, (32, 8, 16, 8, 31)
63, 70, 78, 82, 83, 90, 94
101 1, 2, 3, 4, 7, 21, 22, 25, 29, 36, 46, 48, (34, 4, 20, 4, 33)
49, 66, 68, 71, 79, 83, 84, 92
103 1, 2, 3, 4, 7, 21, 22, 26, 28, 30, 33, 36, 45, (34, 4, 20, 4, 33)
46, 49, 53, 69, 73, 75, 95
1, 2, 3, 4, 7, 21, 22, 25, 33, 35, 46, 47, (34, 2, 24, 2, 33)
53, 56, 66, 74, 75, 81, 88, 89
105 1, 2, 3, 4, 7, 16, 28, 37, 45, 49, 52, 59, (34, 4, 20, 4, 33)
62, 63, 70, 71, 72, 77, 79, 89
129 1, 2, 3, 4, 7, 9, 16, 32, 36, 45, 49, 52, (30, 8, 20, 8, 29)
54, 58, 64, 70, 71, 72, 89, 90
1, 2, 3, 4, 7, 9, 16, 23, 32, 36, 47, 48, 49, (30, 7, 22, 7, 29)
52, 54, 58, 64, 70, 73, 90
1, 2, 3, 4, 7, 9, 13, 21, 22, 27, 32, 46, 47, (34, 4, 20, 4, 33)
52, 58, 66, 72, 74, 88, 90
1, 2, 3, 4, 7, 9, 12, 13, 22, 31, 32, 39, 41, (34, 2, 24, 2, 33)
48, 56, 67, 71, 76, 90, 95
1, 2, 3, 4, 7, 8, 21, 22, 29, 35, 41, 46, (34, 3, 22, 3, 33)
62, 66, 72, 73, 74, 79, 82, 96
130 1, 2, 3, 4, 7, 9, 16, 33, 35, 45, 50, 52, (32, 4, 24, 4, 31)
53, 57, 65, 70, 71, 72, 89, 90
1, 2, 3, 4, 7, 9, 16, 23, 32, 35, 44, 47, 48, (32, 3, 26, 3, 31)
50, 52, 54, 73, 75, 81, 85
131 1, 2, 3, 4, 7, 9, 21, 22, 36, 46, 49, 52, (32, 2, 28, 2, 31)
55, 59, 61, 68, 72, 73, 87, 96
1, 2, 3, 4, 7, 9, 21, 22, 35, 46, 50, 52, 59, (34, 4, 20, 4, 33)
62, 68, 71, 73, 77, 87, 93

48, 55, 71, 82, 84, 92, 95
1, 2, 3, 4, 7, 10, 11, 12, 19, 31, 40, 51, (32, 4, 24, 4, 31)
53, 58, 61, 63, 64, 80, 94, 96
1, 2, 3, 4, 7, 9, 21, 22, 24, 25, 35, 46, 59, (34, 2, 24, 2, 33)
62, 68, 71, 73, 79, 87, 92
1, 2, 3, 4, 7, 9, 21, 22, 24, 25, 35, 46, 47, (32, 8, 16, 8, 31)
59, 62, 66, 79, 88, 89, 92
143 1, 2, 3, 4, 7, 8, 19, 28, 36, 39, 41, 68, 69, (30, 8, 20, 8, 29)
74, 80, 83, 84, 85, 88, 91
1, 2, 3, 4, 7, 8, 19, 23, 28, 36, 39, 47, (30, 7, 22, 7, 29)
48, 73, 74, 80, 83, 84, 85, 91
1, 2, 3, 4, 7, 8, 16, 24, 29, 30, 36, 43, (34, 4, 20, 4, 33)
59, 60, 76, 78, 82, 83, 85, 94
the electronic journal of combinatorics 14 (2007), #R8 11
144 1, 2, 3, 4, 7, 10, 11, 12, 19, 31, 40, 51, (30, 2, 32, 2, 29)
53, 58, 61, 63, 64, 80, 94, 96
1, 2, 3, 4, 7, 9, 21, 22, 35, 46, 50, 52, 59, (34, 2, 24, 2, 33)
62, 68, 71, 73, 77, 87, 93
145 1, 2, 3, 4, 7, 9, 16, 32, 35, 44, 50, 52, 54, (34, 4, 20, 4, 33)
59, 75, 81, 82, 83, 85, 94
146 1, 2, 3, 4, 7, 21, 22, 25, 46, 47, 49, 56, 62, (34, 3, 22, 3, 33)
63, 79, 87, 88, 89, 94, 96
1, 2, 3, 4, 7, 21, 22, 24, 25, 36, 46, 56, (34, 4, 20, 4, 33)
59, 62, 68, 72, 73, 79, 87, 91
1, 2, 3, 4, 7, 12, 25, 37, 40, 43, 49, 56, 58, (32, 4, 24, 4, 31)
63, 66, 76, 78, 86, 87, 95
1, 2, 3, 4, 7, 10, 11, 12, 31, 34, 37, 40, (32, 3, 26, 3, 31)
43, 53, 63, 75, 81, 83, 92, 93
147 1, 2, 3, 4, 7, 9, 21, 22, 24, 25, 35, 46, 47, (32, 4, 24, 4, 31)
59, 62, 66, 79, 88, 89, 92

58, 64, 66, 78, 86, 87, 90, 95
1, 2, 3, 4, 7, 10, 13, 27, 29, 40, 43, 56, 57, (28, 6, 28, 6, 27)
58, 77, 78, 82, 85, 92, 94
1, 2, 3, 4, 7, 10, 13, 27, 29, 40, 43, 56, 57, (28, 2, 36, 2, 27)
58, 59, 78, 83, 85, 91, 96
1, 2, 3, 4, 7, 10, 13, 22, 29, 39, 55, 56, (30, 6, 24, 6, 29)
66, 67, 68, 72, 73, 82, 83, 93
1, 2, 3, 4, 7, 10, 13, 22, 27, 29, 39, 41, 48, (32, 0, 32, 0, 31)
56, 57, 58, 67, 78, 86, 89
1, 2, 3, 4, 7, 10, 12, 13, 29, 33, 40, 43, (28, 0, 40, 0, 27)
53, 56, 59, 78, 79, 80, 85, 94
1, 2, 3, 4, 7, 10, 12, 13, 29, 33, 40, 43, 53, (28, 4, 32, 4, 27)
55, 56, 78, 82, 83, 85, 93
161 1, 2, 3, 4, 5, 8, 17, 29, 31, 33, 37, 53, (32, 4, 24, 4, 31)
60, 61, 64, 69, 76, 77, 83, 84
1, 2, 3, 4, 5, 8, 21, 22, 29, 37, 46, 48, (34, 4, 20, 4, 33)
52, 68, 71, 74, 77, 81, 84, 95
1, 2, 3, 4, 5, 8, 17, 20, 29, 33, 37, 46, 47, (32, 3, 26, 3, 31)
53, 64, 69, 73, 76, 77, 83
162 1, 2, 3, 4, 5, 10, 13, 22, 29, 30, 32, 34, (32, 2, 28, 2, 31)
39, 42, 67, 69, 73, 76, 86, 92
1, 2, 3, 4, 5, 10, 13, 22, 29, 30, 32, 34, 38, (32, 0, 32, 0, 31)
39, 47, 69, 70, 76, 92, 95
1, 2, 3, 4, 5, 8, 17, 30, 32, 36, 38, 54, 65, (30, 8, 20, 8, 29)
67, 68, 69, 75, 77, 83, 88
1, 2, 3, 4, 5, 8, 17, 20, 30, 32, 36, 46, (30, 7, 22, 7, 29)
47, 54, 65, 69, 73, 75, 77, 83
164 1, 2, 3, 4, 5, 10, 13, 27, 39, 44, 50, 57, (34, 4, 20, 4, 33)
59, 63, 76, 80, 85, 86, 87, 95
1, 2, 3, 4, 5, 8, 21, 22, 30, 36, 38, 42, 47, (32, 8, 16, 8, 31)

1, 2, 3, 4, 5, 8, 21, 22, 30, 42, 47, 48, (34, 3, 22, 3, 33)
50, 52, 60, 66, 81, 86, 88, 96
1, 2, 3, 4, 5, 8, 21, 22, 29, 37, 43, 48, 52, (36, 3, 18, 3, 35)
58, 60, 65, 68, 73, 75, 77
169 1, 2, 3, 4, 5, 8, 14, 24, 30, 33, 37, 54, (32, 8, 16, 8, 31)
56, 64, 72, 75, 81, 82, 83, 94
170 1, 2, 3, 4, 5, 13, 21, 22, 27, 47, 48, 50, (34, 4, 20, 4, 33)
52, 59, 63, 70, 74, 84, 88, 94
1, 2, 3, 4, 5, 10, 13, 27, 39, 44, 50, 57, (32, 4, 24, 4, 31)
59, 63, 76, 80, 85, 86, 87, 95
1, 2, 3, 4, 5, 10, 13, 27, 38, 39, 44, 50, 57, (32, 2, 28, 2, 31)
59, 67, 68, 76, 80, 87, 88
1, 2, 3, 4, 5, 9, 21, 22, 36, 47, 48, 51, 55, (32, 8, 16, 8, 31)
56, 70, 78, 84, 85, 88, 96
1, 2, 3, 4, 5, 9, 13, 21, 22, 48, 52, 55, 56, (34, 2, 24, 2, 33)
61, 63, 68, 70, 73, 77, 91
172 1, 2, 3, 4, 5, 8, 21, 22, 36, 42, 48, 51, 52, (34, 4, 20, 4, 33)
59, 60, 63, 68, 73, 81, 90
173 1, 2, 3, 4, 5, 13, 21, 22, 30, 46, 48, 50, (34, 4, 20, 4, 33)
55, 61, 82, 85, 88, 89, 93, 96
1, 2, 3, 4, 5, 8, 17, 24, 33, 39, 52, 57, (32, 8, 16, 8, 31)
59, 65, 66, 78, 79, 80, 92, 93
the electronic journal of combinatorics 14 (2007), #R8 14
174 1, 2, 3, 4, 5, 8, 21, 22, 24, 31, 42, 47, 48, (32, 4, 24, 4, 31)
59, 66, 78, 81, 86, 88, 96
1, 2, 3, 4, 5, 6, 7, 14, 23, 28, 33, 46, 51, (34, 4, 20, 4, 33)
53, 56, 72, 79, 85, 87, 92
1, 2, 3, 4, 5, 6, 7, 10, 20, 23, 39, 44, (34, 1, 26, 1, 33)
50, 57, 63, 76, 79, 80, 87, 94
175 1, 2, 3, 4, 5, 8, 9, 21, 22, 37, 42, 48, 51, (34, 4, 20, 4, 33)

53, 72, 76, 81, 85, 86, 88
218 1, 2, 3, 6, 7, 9, 10, 12, 15, 19, 32, 39, 51, (32, 4, 24, 4, 31)
56, 69, 72, 73, 85, 86, 90
1, 2, 3, 4, 7, 10, 13, 26, 34, 39, 44, 52, (32, 2, 28, 2, 31)
66, 68, 84, 85, 87, 88, 91, 92
the electronic journal of combinatorics 14 (2007), #R8 15
1, 2, 3, 4, 7, 10, 11, 13, 31, 34, 39, 41, (32, 0, 32, 0, 31)
44, 48, 71, 82, 85, 92, 93, 95
1, 2, 3, 4, 7, 8, 10, 22, 30, 32, 33, 36, 41, (34, 2, 24, 2, 33)
52, 57, 64, 72, 73, 85, 95
1, 2, 3, 4, 7, 8, 10, 19, 21, 23, 36, 48, 52, (32, 3, 26, 3, 31)
54, 58, 60, 70, 75, 86, 95
219 1, 2, 3, 5, 7, 11, 14, 19, 23, 28, 32, 56, (32, 0, 32, 0, 31)
67, 73, 75, 78, 81, 85, 86, 87
1, 2, 3, 4, 7, 10, 12, 13, 22, 26, 43, 52, 53, (30, 4, 28, 4, 29)
58, 61, 62, 70, 77, 90, 96
220 1, 2, 3, 4, 6, 10, 11, 13, 27, 38, 44, 53, (32, 2, 28, 2, 31)
56, 59, 61, 66, 67, 71, 72, 88
1, 2, 3, 4, 6, 10, 11, 12, 13, 32, 38, 44, (32, 0, 32, 0, 31)
45, 46, 55, 69, 74, 82, 86, 95
221 1, 2, 3, 4, 6, 9, 16, 20, 25, 26, 46, 53, 58, (30, 4, 28, 4, 29)
59, 72, 74, 77, 79, 82, 84
1, 2, 3, 4, 6, 9, 16, 20, 22, 25, 26, 53, 59, (32, 0, 32, 0, 31)
62, 74, 77, 79, 81, 82, 89
1, 2, 3, 4, 6, 9, 14, 15, 22, 31, 34, 37, 41, (30, 1, 34, 1, 29)
47, 64, 70, 74, 83, 84, 91
1, 2, 3, 4, 6, 8, 10, 18, 34, 39, 51, 52, 54, (32, 4, 24, 4, 31)
61, 63, 66, 80, 90, 91, 96
1, 2, 3, 4, 6, 8, 10, 18, 22, 34, 39, 48, 51, (32, 3, 26, 3, 31)
52, 61, 63, 66, 70, 89, 91

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the electronic journal of combinatorics 14 (2007), #R8 17