Table 4: Partial results in groups of order 𝑣 by Criterion 2. 𝐶 | 𝐻 | = 𝑥 , 𝑦 𝑥 𝑞 = 𝑦 2 = [ 𝑥 , 𝑦 ] and parameters with asterisk indicate new results. No difference set image in 𝐻 implies no diff. set image in Dihedral group of same order. ? means the number of groups of order 𝑣 is unknown.

( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐻 | Factoring of 𝑝 in [ 𝜁 | 𝐻 / 𝑔 | ] No. of groups of order 𝑣 No. of groups ruled outSolutions in 𝐻

1(78, 22, 6) 4 2 26 2 6 1 ( m o d 1 3 ) 64 4 + 2 𝑥 in 𝐻 / 𝑔
2(204, 29, 4) 5 5 34 5 8 1 ( m o d 1 7 ) 1210 5 + 2 𝑥 in 𝐻 / 𝑔
3 (1140, 68, 4) 8 2 38 2 9 1 ( m o d 1 9 ) 4129 8 + 4 𝑥 in 𝐻 / 𝑔
4 (396, 80, 16) 8 2 66 2 5 1 ( m o d 𝑏 )
𝑏 = 1 1 , 3 3
3015None in 𝐻 / 𝑔
5 (300, 92, 48) 8 2 50 2 1 0 1 ( m o d 2 5 ) 499 8 + 4 𝑥 in 𝐻 / 𝑔
6 (980, 89, 8) 9 3 14 3 3 1 ( m o d 7 ) 3431 9 + 7 𝑥 𝑦
7 (456, 105, 24) 9 3 38 3 9 1 ( m o d 1 9 ) 5439 9 + 6 𝑥 in 𝐻 / 𝑔
8 (1856, 106, 6) 10 2, 5 58 𝑎 1 ( m o d 2 9 )
𝑎 = 2 1 4 , 5 7
16301387 1 0 + 4 𝑥 in 𝐻 / 𝑔
9 (7504, 123, 2) 11 11 134 1 1 3 3 1 ( m o d 6 7 ) ?? 1 1 + 2 𝑥 in 𝐻 / 𝑔
1 0 (3876, 125, 4) 11 11 34 1 1 8 1 ( m o d 1 7 ) 4034 1 1 + 8 𝑥 in 𝐻 / 𝑔
1 1 (2870, 152, 8) 12 2, 3 82 𝑎 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4
128 1 2 + 4 𝑥 in 𝐻 / 𝑔
1 2 (666, 210, 66) 12 2, 3 74 𝑎 1 ( m o d 3 7 )
𝑎 = 2 1 8 , 3 9
1810 1 2 + 6 𝑥 in 𝐻 / 𝑔
1 3 (610, 232, 88) 12 2, 3 122 𝑎 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 3 5
64 1 2 + 4 𝑥 in 𝐻 / 𝑔
1 4 (14536, 171, 2) 13 13 46Remark 2.3?? 1 3 + 8 𝑥 in 𝐻 / 𝑔
1 5 (7440, 173, 4) 13 13 62 1 3 1 5 1 ( m o d 3 1 ) ?? 1 3 + 6 𝑥 in 𝐻 / 𝑔
1 6 (5076, 175, 6) 13 13 94 1 3 2 3 1 ( m o d 4 7 ) ?? 1 3 + 4 𝑥 in 𝐻 / 𝑔
1 7 (2716, 181, 12) 13 13 194 1 3 4 8 1 ( m o d 9 7 ) 11 9 1 3 + 2 𝑥 in 𝐻 / 𝑔
1 8 (19504, 198, 2) 14 2, 7 106 𝑎 1 ( m o d 5 3 )
𝑎 = 2 2 6 , 7 1 3
?? 1 4 + 4 𝑥 in 𝐻 / 𝑔
1 9 (1696, 226, 30) 14 2, 7 212 2 2 6 1 ( m o d 5 3 )
7 1 3 1 ( m o d 𝑏 )
𝑏 = 5 3 , 1 0 6 , 2 1 2
235194 1 4 + 𝑥 𝑦
2 0 (8856, 231, 6) 15 3, 5 82 𝑎 1 ( m o d 4 1 )
𝑎 = 3 4 , 5 2 0
?? 1 5 + 6 𝑥 in 𝐻 / 𝑔
2 1 (1508, 275, 50) 15 3, 5 58 𝑎 1 ( m o d 2 9 )
𝑎 = 3 1 4 , 5 7
15 11 1 5 + 1 0 𝑥 in 𝐻 / 𝑔
2 2 (976, 351, 126) 15 3, 5 122 𝑎 1 ( m o d 6 1 )
𝑎 = 3 5 , 5 1 5
51 14 1 5 + 6 𝑥 in 𝐻 / 𝑔
2 3 (3256, 280, 24) 16 2 74 2 1 8 1 ( m o d 3 7 ) ?? 1 6 + 8 𝑥 in 𝐻 / 𝑔
2 4 (1036, 460, 204) 16 2 74 2 1 8 1 ( m o d 3 7 ) 119 1 6 + 6 𝑥 𝑦 ; 6 𝑥 𝑦 + 8 ( 1 + 𝑥 + 𝑦 𝑥 𝑦 )
2 5 (21390, 293, 4) 17 17 62 1 7 1 5 1 ( m o d 3 1 ) 3616 1 7 + 1 0 𝑥 in 𝐻 / 𝑔
2 6 (52976, 326, 2) 18 2, 3 86 3 2 1 1 ( m o d 4 3 ) See Remark 2.2 for 2 ?? 1 8 + 8 𝑥 in 𝐻 / 𝑔
2 7 (18096, 330, 6) 18 2, 3 58 𝑎 1 4 1 ( m o d 2 9 )
𝑎 = 2 , 3
?? 1 8 + 1 2 𝑥 in 𝐻 / 𝑔
2 8 (1776, 426, 102) 18 2, 3 74 𝑎 1 ( m o d 3 7 )
𝑎 = 2 1 8 , 3 9
260170 1 8 + 1 2 𝑥 in 𝐻 / 𝑔
2 9 (65704, 363, 2) 19 19 382 1 9 9 5 1 ( m o d 1 9 1 ) ? ? 1 9 + 2 𝑥 in 𝐻 / 𝑔
3 0 (22388, 367, 6) 19 19 386 1 9 9 6 1 ( m o d 1 9 3 ) 15 11 1 9 + 2 𝑥 in 𝐻 / 𝑔
3 1 (13728, 371, 10) 19 19 26 1 9 6 1 ( m o d 1 3 ) ? ? 1 9 + 1 5 𝑥 𝑦
3 2 (7960, 379, 18) 19 19 398 1 9 9 1 ( m o d 1 9 9 ) ?? 1 9 + 2 𝑥 in 𝐻 / 𝑔
3 3 (5084, 391, 30) 19 19 82 1 9 2 0 1 ( m o d 4 1 ) 11 9 1 9 + 1 0 𝑥 in 𝐻 / 𝑔
3 4 (2256, 451, 90) 19 19 94 1 9 2 3 1 ( m o d 4 7 ) ? ? 1 9 + 1 0 𝑥 in 𝐻 / 𝑔
3 5 (40704, 404, 4) 20 2, 5 106 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 2 , 5
?? 2 0 + 8 𝑥 in 𝐻 / 𝑔
3 6 (16770, 410, 10) 20 2, 5 86 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 5 2 1
48 24 2 0 + 1 0 𝑥 in 𝐻 / 𝑔
3 7 (1830, 590, 190) 20 2, 5 122 𝑎 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 5 1 5
18 8 2 0 + 1 0 𝑥 i n 𝐻 / 𝑔
3 8 (49369, 445, 4) 21 3, 7 466 𝑎 1 ( m o d 2 3 3 )
𝑎 = 3 1 1 6 , 7 5 8
1511 2 1 + 2 𝑥 in 𝐻 / 𝑔
3 9 (17064, 453, 12) 21 3, 7 158 𝑎 3 9 1 ( m o d 7 9 )
𝑎 = 3 , 7
?? 2 1 + 6 𝑥 in 𝐻 / 𝑔
4 0 (14756, 455, 14) 21 3, 7 34 𝑎 8 1 ( m o d 1 7 )
𝑎 = 3 , 7
2723 2 1 + 1 4 𝑥 𝑦
4 1 (10604, 461, 20) 21 3, 7 482 𝑎 1 ( m o d 2 4 1 )
𝑎 = 3 6 0 , 7 1 2 0
119 2 1 + 2 𝑥 in 𝐻 / 𝑔
4 2 (7380, 471, 30) 21 3, 7 82 𝑎 1 ( m o d 4 1 )
𝑎 = 3 4 , 7 2 0
14989 2 1 + 1 2 𝑥 in 𝐻 / 𝑔
4 3 (5336, 485, 44) 21 3, 7 46Remark 2.3? ? 2 1 + 1 1 𝑥 𝑦
4 4 (3128, 531, 90) 21 3, 7 46Remark 2.3? ? 2 1 + 1 2 𝑥 𝑦
4 5 (2408, 581, 140) 21 3, 7 86 𝑎 1 ( m o d 4 3 )
𝑎 = 3 2 1 , 7 3
? ? 2 1 + 1 4 𝑥 in 𝐻 / 𝑔
4 6 (70890, 533, 4) 23 23 278 2 3 2 3 1 ( m o d 1 3 9 ) 24 16 2 3 + 4 𝑥 in 𝐻 / 𝑔
4 7 (47616, 535, 8) 23 23 62 2 3 5 1 ( m o d 3 1 ) ? ? 2 3 + 1 8 𝑥 in 𝐻 / 𝑔
4 8 (13776, 551, 22) 23 23 82 2 3 5 1 ( m o d 4 1 ) ? ? 2 3 + 7 𝑥 𝑦
4 9 (166754, 578, 2) 24 2, 3 86 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 3 2 1
12 8 2 4 + 1 4 𝑥 in 𝐻 / 𝑔
5 0 (56358, 582, 6) 24 2, 3 202 𝑎 5 0 1 ( m o d 1 0 1 )
𝑎 = 2 , 3
? ? 2 4 + 6 𝑥 in 𝐻 / 𝑔
5 1 (10388, 612, 36) 24 2, 3 106 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 2 , 3
34 28 2 4 + 1 2 𝑥 in 𝐻 / 𝑔
5 2 (4844, 668, 92) 24 2, 3 346 𝑎 8 6 1 ( m o d 1 7 3 )
𝑎 = 2 , 3
11 9 2 4 + 4 𝑥 in 𝐻 / 𝑔
5 3 (3690, 714, 138) 24 2, 3 82 𝑎 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4
30 20 2 4 + 1 8 𝑥 in 𝐻 / 𝑔
5 4 (3172, 756, 180) 24 2, 3 122 𝑎 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 3 5
15 11 2 4 + 6 𝑥 𝑦 , 6 𝑥 𝑦 + 1 2 ( 1 + 𝑥 + 𝑦 𝑥 𝑦 )
5 5 (2332, 1036, 460) 24 2, 3 106 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 2 , 3
11 9 2 4 + 2 0 𝑥 in 𝐻 / 𝑔
5 6 (196252, 627, 2) 25 5326 5 2 7 1 ( m o d 1 6 3 ) ? ? 2 5 + 4 𝑥 in 𝐻 / 𝑔
5 7 (66256, 631, 6) 25 582 5 1 0 1 ( m o d 4 1 ) ? ? 2 5 + 1 6 𝑥 in 𝐻 / 𝑔
5 8 (50008, 633, 8) 25 594 5 2 3 1 ( m o d 4 7 ) ? ? 2 5 + 1 4 𝑥 in 𝐻 / 𝑔
5 9 (17524, 649, 24) 25 5674 5 5 6 1 ( m o d 3 3 7 ) 15 11 2 5 + 2 𝑥 𝑦
6 0 (14280, 655, 30) 25 534 5 8 1 ( m o d 1 7 ) ? ? 2 5 + 2 0 𝑥 𝑦
6 1 (11040, 665, 40) 25 512 5 1 ( m o d 𝑏 ) 𝑏 = 2 , 3 ? ? 2 5 + 1 1 5 𝑥 in 𝐻 / 𝑔
6 2 (11040, 665, 40) 25 546Remark 2.3 ? ? 2 5 + 1 5 𝑥 𝑦
6 3 (5104, 729, 104) 25 558 5 7 1 ( m o d 2 9 ) ? ? 2 5 + 1 3 𝑥 𝑦
6 4 (2812, 937, 312) 25 574 5 1 8 1 ( m o d 3 7 ) 11 9 2 5 + 1 3 𝑥 𝑦