Table 1: Parameter sets that do not exist by Criterion 1. 𝐶 | 𝐺 / 𝑁 | = 𝑥 and parameters with asterisk indicate new results.

( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐺 / 𝑁 | Factoring of 𝑝 in [ 𝜁 | 𝐺 / 𝑁 | ] No. of groups of order 𝑣 Solutions in 𝐺 / 𝑁

1(115, 19, 3)4 223 Remark 2.31 4 + 𝑥
2(1333, 37, 1) 6 2, 3 43 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 3 2 1
1 6 + 𝑥
3(221, 45, 9) 6 2, 3 17 𝑎 1 ( m o d 1 7 )
𝑎 = 2 4 , 3 8
1 6 + 3 𝑥
4(145, 64, 28) 6 2, 3 29 𝑎 1 4 1 ( m o d 2 9 ) ,
𝑎 = 2 , 3
1 6 + 2 𝑥
5(1463, 86, 5) 9 3 19 3 9 1 ( m o d 1 9 ) 1 9 + 5 𝑥
6(583, 97, 16) 9 3 53 3 2 6 1 ( m o d 5 3 ) 1 9 + 2 𝑥
7(345, 129, 48) 9 3 23Remark 2.3 1 9 + 6 𝑥
8(3503, 103, 3) 10 2, 5 113 𝑎 1 ( m o d 1 1 3 )
𝑎 = 2 1 4 , 5 5 6
1 1 0 + 𝑥
9(2185, 105, 5) 10 2, 5 23Remark 2.3 1 1 0 + 5 𝑥
10(1309, 109, 9) 10 2, 5 17 𝑎 1 ( m o d 1 7 )
𝑎 = 2 4 , 5 8
1 1 0 + 7 𝑥
11(1037, 112, 12) 10 2, 5 61 𝑎 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 5 1 5
1 1 0 + 2 𝑥
1 2 (621, 125, 25) 10 2, 5 69 5 1 1 1 ( m o d 𝑏 )
𝑏 = 2 3 , 6 9
2 factors trivially in [ 𝜁 𝑎 ] , 𝑎 = 2 3 , 6 9
5 1 0 + 5 𝑥 in 𝐶 2 3 ; None in 𝐶 6 9
13(469, 144, 44) 10 2, 5 67 𝑎 1 ( m o d 6 7 )
𝑎 = 2 3 3 , 5 1 1
1 1 0 + 2 𝑥
14(407, 175, 75) 10 2, 5 37 𝑎 1 8 1 ( m o d 3 7 )
𝑎 = 2 , 5
1 1 0 + 5 𝑥
15(3151, 126, 5) 11 11 137 1 1 3 4 1 ( m o d 1 3 7 ) 1 1 1 + 𝑥
1 6 (483, 241, 120) 11 11 23 1 1 1 1 1 ( m o d 𝑏 )
𝑏 = 2 3 , 6 9
2 1 1 + 1 0 𝑥 in 𝐶 2 3 ; None in 𝐶 6 9
17(561, 176, 55) 11 11 187 1 1 8 1 ( m o d 1 7 )
11 factors trivially in [ 𝜁 𝑎 ] , 𝑎 = 1 7 , 1 8 7
1 1 1 + 1 1 𝑥 in 𝐶 1 7 ; None in 𝐶 1 8 7
18(20881, 145, 1) 12 2, 3 157 𝑎 1 ( m o d 1 5 7 )
𝑎 = 2 2 6 , 3 3 9
1 1 2 + 𝑥
19(1591, 160, 16) 12 2, 3 43 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 3 2 1
1 1 2 + 4 𝑥
20(9805, 172, 3)13 1337 1 3 1 8 1 ( m o d 3 7 ) 1 1 3 + 5 𝑥
2 1 (3895, 177, 8)13 1319 1 3 9 1 ( m o d 1 9 ) 2 1 3 + 1 0 𝑥
2 2 (1711, 190, 21) 13 13 29 1 3 7 1 ( m o d 2 9 ) 2 1 3 + 7 𝑥
23(2323, 216, 20) 14 14 23 1 4 1 1 1 ( m o d 2 3 ) 1 1 4 + 1 0 𝑥
2 4 (9951, 200, 4) 14 2, 7 107 𝑎 5 3 1 ( m o d 1 0 7 )
𝑎 = 2 , 7
2 1 4 + 2 𝑥
25(8041, 201, 5) 14 2, 7 43 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 7 3
1 1 4 + 5 𝑥
26(793, 352, 156) 14 2, 7 61 𝑎 3 0 1 ( m o d 6 1 )
𝑎 = 2 , 7
1 1 4 + 6 𝑥
27(50851, 226, 1) 15 3, 5 241 𝑎 1 ( m o d 2 4 1 )
𝑎 = 3 6 0 , 5 2 0
1 1 5 + 𝑥
2 8 (2871, 246, 21) 15 3, 5 29 𝑎 1 ( m o d 2 9 )
𝑎 = 3 1 4 , 5 7
2 1 5 + 9 𝑥
29(2491, 250, 25) 15 3, 5 53 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 3 , 5
1 1 5 + 5 𝑥
3 0 (13573, 261, 5) 16 2 277 2 4 6 1 ( m o d 2 7 7 ) 2 1 6 + 𝑥
31(4879, 271, 15) 16 2 41 2 1 0 1 ( m o d 4 1 ) 1 1 6 + 7 𝑥
3 2 (26815, 328, 4) 18 2, 3 173 𝑎 8 6 1 ( m o d 1 7 3 )
𝑎 = 2 , 3
2 1 8 + 2 𝑥
3 3 (4551, 351, 27) 18 2, 3 41 𝑎 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4
2 1 8 + 9 𝑥
3 4 (16975, 369, 8) 19 19 97 1 9 1 6 1 ( m o d 9 7 ) 2 1 9 + 4 𝑥
3 5 (15171, 370, 9) 19 19 389 1 9 9 7 1 ( m o d 3 8 9 ) 2 1 9 + 𝑥
36(2599, 433, 72) 19 19 113 1 9 5 6 1 ( m o d 1 1 3 ) 1 1 9 + 4 𝑥
37(11455, 415, 15) 20 2, 5 29 𝑎 1 ( m o d 2 9 )
𝑎 = 2 1 4 , 5 7
1 2 0 + 1 5 𝑥
38(3657, 457, 57) 20 2, 5 53 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 2 , 5
1 2 0 + 9 𝑥
39 (194923, 442, 1) 21 3, 7 463 𝑎 1 ( m o d 4 6 3 )
𝑎 = 3 2 3 1 , 7 7 7
1 2 1 + 𝑥
40 (28609, 448, 7) 21 3, 7 67 𝑎 1 ( m o d 6 7 )
𝑎 = 3 1 1 , 7 3 3
1 2 1 + 7 𝑥
4 1 (18533, 452, 11) 21 3, 7 43 𝑎 1 ( m o d 4 3 )
𝑎 = 3 2 1 , 7 3
2 2 1 + 1 1 𝑥
4 2 (13833, 456, 15) 21 3, 7 53 𝑎 1 ( m o d 5 3 )
𝑎 = 3 2 6 , 7 1 3
2 2 1 + 9 𝑥
43(4891, 490, 49) 21 3, 7 73 𝑎 1 ( m o d 7 3 )
𝑎 = 3 6 , 7 1 2
1 2 1 + 7 𝑥
44(3649, 513, 72) 21 3, 7 89 𝑎 4 4 1 ( m o d 8 9 )
𝑎 = 3 , 7
1 2 1 + 6 𝑥
45(2941, 540, 99) 21 3, 7 173 𝑎 8 6 1 ( m o d 1 7 3 )
𝑎 = 3 , 7
1 2 1 + 3 𝑥
46(1919, 686, 245) 21 3, 7 101 𝑎 5 0 1 ( m o d 1 0 1 )
𝑎 = 3 , 7
1 2 1 + 7 𝑥
47(1769, 833, 392) 21 3, 7 61 𝑎 1 ( m o d 6 1 )
𝑎 = 3 5 , 7 3 0
1 2 1 + 1 4 𝑥
4 8 (78895, 487, 3) 22 2, 11 509 𝑎 1 ( m o d 5 0 9 )
𝑎 = 2 2 5 4 , 1 1 1 2 7
2 2 2 + 𝑥
49(20461, 496, 12) 22 2, 11 37 𝑎 1 ( m o d 3 7 )
𝑎 = 2 1 8 , 1 1 3
1 2 2 + 1 4 𝑥
50(4081, 561, 77) 22 2, 11 53 𝑎 1 ( m o d 5 3 )
𝑎 = 2 2 6 , 1 1 1 3
1 2 2 + 1 1 𝑥
51(3835, 568, 84) 22 2, 11 59 𝑎 2 9 1 ( m o d 5 9 )
𝑎 = 2 , 1 1
1 2 2 + 1 0 𝑥
52(3601, 576, 92) 22 2, 11 277 𝑎 1 ( m o d 2 7 7 )
𝑎 = 2 4 6 , 1 1 1 3 8
1 2 2 + 2 𝑥
53(94165, 532, 3)23 23 37 2 3 6 1 ( m o d 3 7 ) 1 2 3 + 1 5 𝑥
5 4 (18531, 545, 16)23 23 71 2 3 7 1 ( m o d 7 1 ) 2 2 3 + 8 𝑥
55(8557, 621, 45)24 2, 3 43 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 3 2 1
1 2 4 + 1 5 𝑥
56(2959, 783, 207)24 2, 3 269 𝑎 1 3 4 1 ( m o d 2 6 9 )
𝑎 = 2 , 3
1 2 4 + 3 𝑥
5 7 (131253, 628, 3)25 5 653 5 3 2 6 1 ( m o d 6 5 3 ) 2 2 5 + 𝑥
58(11289, 664, 39)25 5 53 5 2 6 1 ( m o d 5 3 ) 1 2 5 + 1 3 𝑥
59(6205, 705, 80)25 5 73 5 3 6 1 ( m o d 7 3 ) 1 2 5 + 1 0 𝑥
60(3115, 865, 240)25 5 89 5 2 2 1 ( m o d 8 9 ) 1 2 5 + 1 0 𝑥