Table 2: Partial results in groups of order by Criterion 1. and parameters with asterisk indicate new results. ? means the number of groups of order is unknown.
 ( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐺 / 𝑁 | Factoring of 𝑝 in ℤ [ 𝜁 | 𝐺 / 𝑁 | ] No. of groups of order 𝑣 No. of groups ruled out Solutions in 𝐺 / 𝑁 1 (171, 51, 15) 6 2,3 19 𝑎 9 ≡ − 1 ( m o d 1 9 ) 𝑎 = 2 , 3 5 2 − 6 + 3 ⟨ 𝑥 ⟩ 2 (155, 56, 20) 6 2,3 31 3 1 5 ≡ − 1 ( m o d 3 1 ) 2 factors trivially [20] 2 1 − 6 + 2 ⟨ 𝑥 ⟩ 3 (231, 70, 21) 7 7 77 7 5 ≡ − 1 ( m o d 1 1 ) 7 factors trivially in ℤ [ 𝜁 𝑏 ] , 𝑏 = 1 1 , 7 7 2 1 − 7 + 7 ⟨ 𝑥 ⟩ in 𝐶 7 ; None in 𝐶 7 7 4 ∗ (2325, 84, 3) 9 3 31 3 1 5 ≡ − 1 ( m o d 3 1 ) 10 3 − 9 + 3 ⟨ 𝑥 ⟩ , 5 ∗ (10101, 101, 1) 10 2,5 37 𝑎 1 8 ≡ − 1 ( m o d 3 7 ) 𝑎 = 2 , 3 14 5 − 1 0 + 3 ⟨ 𝑥 ⟩ 6 (715, 154, 33) 11 11 143 1 1 6 ≡ − 1 ( m o d 1 3 ) 11 factors trivially in ℤ [ 𝜁 𝑏 ] , 𝑏 = 1 3 , 1 4 3 2 1 1 1 + 1 1 ⟨ 𝑥 ⟩ in 𝐶 1 3 ; None in 𝐶 1 4 3 7 ∗ (7155, 147, 3) 12 2,3 53 𝑎 2 6 ≡ − 1 ( m o d 5 3 ) 𝑎 = 2 , 3 ? ? − 1 2 + 3 ⟨ 𝑥 ⟩ 8 ∗ (38613, 197, 1) 14 2,7 211 𝑎 1 0 5 ≡ − 1 ( m o d 2 1 1 ) 𝑎 = 2 , 7 5 2 − 1 4 + ⟨ 𝑥 ⟩ 9 ∗ (5859, 203, 7) 14 2,7 31 7 1 5 ≡ − 1 ( m o d 3 1 ) 2 factors trivially [20] ? ? − 1 4 + 7 ⟨ 𝑥 ⟩ 1 0 ∗ (903, 287, 91) 14 2,7 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 2 7 , 7 3 7 2 − 1 4 + 7 ⟨ 𝑥 ⟩ 1 1 ∗ (2255, 392, 68) 18 2,3 41 𝑎 ≡ − 1 ( m o d 4 1 ) 𝑎 = 2 1 0 , 3 4 7 2 − 1 8 + 1 0 ⟨ 𝑥 ⟩ 1 2 ∗ (160401, 401, 1) 20 2,5 421 𝑎 ≡ − 1 ( m o d 4 2 1 ) 𝑎 = 2 2 1 0 , 5 1 0 5 5 2 − 2 0 + ⟨ 𝑥 ⟩ 1 3 ∗ (23607, 407, 7) 20 2,5 61 𝑎 ≡ − 1 ( m o d 6 1 ) 𝑎 = 2 3 0 , 5 1 5 11 5 − 2 0 + 7 ⟨ 𝑥 ⟩ 1 4 ∗ (22451, 450, 9) 21 3,7 157 𝑎 ≡ − 1 ( m o d 1 5 7 ) 𝑎 = 3 3 9 , 7 2 6 2 1 − 2 1 + 3 ⟨ 𝑥 ⟩ 1 5 ∗ (2619, 561, 120) 21 3,7 97 𝑎 ≡ − 1 ( m o d 9 7 ) 𝑎 = 3 2 4 , 7 4 8 13 5 − 2 1 + 6 ⟨ 𝑥 ⟩ 16 (2211, 715, 231) 22 2,11 67 𝑎 3 3 ≡ − 1 ( m o d 6 7 ) 𝑎 = 2 , 1 1 4 1 − 2 2 + 1 1 ⟨ 𝑥 ⟩ 17 (7450, 573, 44) 23 23 149 2 3 7 4 ≡ − 1 ( m o d 1 4 9 ) 10 5 − 2 3 + 4 ⟨ 𝑥 ⟩ 18 (111555, 579, 3) 24 2,3 67 𝑎 ≡ − 1 ( m o d 6 7 ) 𝑎 = 2 3 3 , 3 1 1 ? ? − 2 4 + 9 ⟨ 𝑥 ⟩ 19 (37961, 585, 9) 24 2,3 29 𝑎 1 4 ≡ − 1 ( m o d 2 9 ) 𝑎 = 2 , 3 2 1 − 2 4 + 2 1 ⟨ 𝑥 ⟩ 20 (23247, 591, 15) 24 2,3 41 𝑎 ≡ − 1 ( m o d 4 1 ) 𝑎 = 2 1 0 , 3 4 ? ? − 2 4 + 1 5 ⟨ 𝑥 ⟩ 21 (25641, 641, 16) 25 5 37 5 1 8 ≡ − 1 ( m o d 3 7 ) 14 4 − 2 5 + 1 8 ⟨ 𝑥 ⟩