Primitive association schemes with up to 22 and 24 vertices

See Classification of primitive association schemes of order up to 22 (with Izumi Miyamoto).
The following table contains all primitive association schemes with up to 22 and 24 vertices, where association schemes of class 1 and cyclotomic schemes with prime number vertices are omitted. Names of primitive permutation groups are as in GAP library.

number of verticesvalencies primitive permutation groups
9 1, 4, 4 3^2:4, 3^2:D8
10 1, 3, 6 A(5), S(5)
15 1, 6, 8 A(6), S(6)
1, 7, 7 None (prim15)
16 1, 5, 10 (2^4:5).4, 2^4:A_5, 2^4:S_5
1, 6, 9 (A_4xA_4):2, 2^4.3^2:4,
2^4.S_3xS_3, (S_4xS_4):2
1, 6, 9 None (prim16_1)
1, 5, 5, 5 2^4:5, 2^4:D_10
1, 5, 5, 5 None (prim16_2)
19 1, 9, 9 None (prim19)
21 1, 10, 10 A(7), S(7)
1, 4, 8, 8 PGL(2,7)

Note that there exists no non-trivial primitive scheme with 20, 22, or 24 vertices.
For 23 vertices, see here.

Relation matrices of non transitive group schemes are as follows. 
prim15

0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 
2 0 1 1 1 2 2 2 1 1 1 1 2 2 2 
2 2 0 1 1 1 2 2 1 2 2 2 1 1 1 
2 2 2 0 1 2 1 1 2 1 1 2 1 1 2 
2 2 2 2 0 1 1 1 1 1 2 1 2 2 1 
2 1 2 1 2 0 1 2 2 2 1 1 1 2 1 
2 1 1 2 2 2 0 1 1 2 1 2 2 1 1 
2 1 1 2 2 1 2 0 2 1 2 1 1 1 2 
1 2 2 1 2 1 2 1 0 2 1 1 2 1 2 
1 2 1 2 2 1 1 2 1 0 1 2 1 2 2 
1 2 1 2 1 2 2 1 2 2 0 1 1 2 1 
1 2 1 1 2 2 1 2 2 1 2 0 2 1 1 
1 1 2 2 1 2 1 2 1 2 2 1 0 1 2 
1 1 2 2 1 1 2 2 2 1 1 2 2 0 1 
1 1 2 1 2 2 2 1 1 1 2 2 1 2 0 


prim16_1

0 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 
1 0 1 1 2 2 2 1 1 1 2 2 2 2 2 2 
1 1 0 2 1 2 2 1 2 2 1 1 2 2 2 2 
1 1 2 0 2 1 2 2 1 2 2 2 1 1 2 2 
1 2 1 2 0 2 1 2 2 2 1 2 1 2 1 2 
1 2 2 1 2 0 1 2 2 2 2 1 2 1 2 1 
1 2 2 2 1 1 0 2 2 1 2 2 2 2 1 1 
2 1 1 2 2 2 2 0 2 1 2 1 2 1 1 2 
2 1 2 1 2 2 2 2 0 1 1 2 1 2 2 1 
2 1 2 2 2 2 1 1 1 0 2 2 2 2 1 1 
2 2 1 2 1 2 2 2 1 2 0 1 1 2 2 1 
2 2 1 2 2 1 2 1 2 2 1 0 2 1 2 1 
2 2 2 1 1 2 2 2 1 2 1 2 0 1 1 2 
2 2 2 1 2 1 2 1 2 2 2 1 1 0 1 2 
2 2 2 2 1 2 1 1 2 1 2 2 1 1 0 2 
2 2 2 2 2 1 1 2 1 1 1 1 2 2 2 0 


prim16_2

0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 
1 0 2 2 3 3 1 1 2 2 3 1 1 2 3 3 
1 2 0 3 2 3 1 2 1 3 2 2 3 1 1 3 
1 2 3 0 3 2 2 1 3 1 2 3 2 1 3 1 
1 3 2 3 0 2 2 3 2 1 1 1 3 3 1 2 
1 3 3 2 2 0 3 2 1 2 1 3 1 3 2 1 
2 1 1 2 2 3 0 3 3 1 1 2 3 2 3 1 
2 1 2 1 3 2 3 0 1 3 1 3 2 2 1 3 
2 2 1 3 2 1 3 1 0 1 3 1 2 3 2 3 
2 2 3 1 1 2 1 3 1 0 3 2 1 3 3 2 
2 3 2 2 1 1 1 1 3 3 0 3 3 1 2 2 
3 1 2 3 1 3 2 3 1 2 3 0 2 1 2 1 
3 1 3 2 3 1 3 2 2 1 3 2 0 1 1 2 
3 2 1 1 3 3 2 2 3 3 1 1 1 0 2 2 
3 3 1 3 1 2 3 1 2 3 2 2 1 2 0 1 
3 3 3 1 2 1 1 3 3 2 2 1 2 2 1 0


prim19

0 1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2
2 0 1 1 1 1 1 1 1 2 1 2 2 2 2 2 1 2 2
2 2 0 2 1 1 2 1 1 1 2 1 2 2 2 1 2 1 1
2 2 1 0 2 2 1 1 2 2 2 1 1 1 2 1 1 2 1
1 2 2 1 0 1 2 2 1 2 2 2 1 2 1 1 1 2 1
1 2 2 1 2 0 2 1 2 1 1 2 2 1 2 2 1 1 1
1 2 1 2 1 1 0 1 2 2 2 2 1 1 1 2 2 1 2
2 2 2 2 1 2 2 0 1 1 1 1 1 1 1 2 1 2 2
2 2 2 1 2 1 1 2 0 2 1 1 2 1 1 1 2 1 2
2 1 2 1 1 2 1 2 1 0 2 2 1 1 2 2 2 1 1
1 2 1 1 1 2 1 2 2 1 0 1 2 2 1 2 2 2 1
1 1 2 2 1 1 1 2 2 1 2 0 2 1 2 1 1 2 2
1 1 1 2 2 1 2 2 1 2 1 1 0 1 2 2 2 2 1
2 1 1 2 1 2 2 2 2 2 1 2 2 0 1 1 1 1 1
2 1 1 1 2 1 2 2 2 1 2 1 1 2 0 2 1 1 2
2 1 2 2 2 1 1 1 2 1 1 2 1 2 1 0 2 2 1
1 2 1 2 2 2 1 2 1 1 1 2 1 2 2 1 0 1 2
1 1 2 1 1 2 2 1 2 2 1 1 1 2 2 1 2 0 2
1 1 2 2 2 2 1 1 1 2 2 1 2 2 1 2 1 1 0
hanaki@math.shinshu-u.ac.jp

1998/12/21