Table 4: Partial results in groups of order by Criterion 2. and parameters with asterisk indicate new results. No difference set image in implies no diff. set image in Dihedral group of same order. ? means the number of groups of order is unknown.
 ( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐻 | Factoring of 𝑝 in ℤ [ 𝜁 | 𝐻 / ⟨ 𝑔 ⟩ | ] No. of groups of order 𝑣 No. of groups ruled out Solutions in 𝐻 1 (78, 22, 6) 4 2 26 2 6 ≡ − 1 ( m o d 1 3 ) 6 4 − 4 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 2 (204, 29, 4) 5 5 34 5 8 ≡ − 1 ( m o d 1 7 ) 12 10 − 5 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 3 ∗ (1140, 68, 4) 8 2 38 2 9 ≡ − 1 ( m o d 1 9 ) 41 29 − 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 30 15 None in 𝐻 / ⟨ 𝑔 ⟩ 5 ∗ (300, 92, 48) 8 2 50 2 1 0 ≡ − 1 ( m o d 2 5 ) 49 9 − 8 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 6 ∗ (980, 89, 8) 9 3 14 3 3 ≡ − 1 ( m o d 7 ) 34 31 − 9 + 7 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 7 ∗ (456, 105, 24) 9 3 38 3 9 ≡ − 1 ( m o d 1 9 ) 54 39 − 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 1630 1387 − 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 ) 40 34 − 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 12 8 − 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 18 10 − 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 6 4 − 1 2 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 1 4 ∗ (14536, 171, 2) 13 13 46 Remark 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 2 235 194 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 ) 11 9 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 ) 36 16 − 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 9 260 170 − 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 15 11 − 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 27 23 − 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 11 9 − 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 0 149 89 − 2 1 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 4 3 ∗ (5336, 485, 44) 21 3, 7 46 Remark 2.3 ? ? − 2 1 + 1 1 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 4 4 ∗ (3128, 531, 90) 21 3, 7 46 Remark 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 5 326 5 2 7 ≡ − 1 ( m o d 1 6 3 ) ? ? − 2 5 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 5 7 ∗ (66256, 631, 6) 25 5 82 5 1 0 ≡ − 1 ( m o d 4 1 ) ? ? − 2 5 + 1 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 5 8 ∗ (50008, 633, 8) 25 5 94 5 2 3 ≡ − 1 ( m o d 4 7 ) ? ? − 2 5 + 1 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 5 9 ∗ (17524, 649, 24) 25 5 674 5 5 6 ≡ − 1 ( m o d 3 3 7 ) 15 11 − 2 5 + 2 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 6 0 ∗ (14280, 655, 30) 25 5 34 5 8 ≡ − 1 ( m o d 1 7 ) ? ? − 2 5 + 2 0 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 6 1 ∗ (11040, 665, 40) 25 5 12 5 ≡ − 1 ( m o d 𝑏 ) 𝑏 = 2 , 3 ? ? − 2 5 + 1 1 5 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩ 6 2 ∗ (11040, 665, 40) 25 5 46 Remark 2.3 ? ? − 2 5 + 1 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 6 3 ∗ (5104, 729, 104) 25 5 58 5 7 ≡ − 1 ( m o d 2 9 ) ? ? − 2 5 + 1 3 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ 6 4 ∗ (2812, 937, 312) 25 5 74 5 1 8 ≡ − 1 ( m o d 3 7 ) 11 9 − 2 5 + 1 3 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩