Cross-Reference List of Bol Loops

I have been working on classifying Bol loops of small order. Here I have listed some names of loops as they appear in other sources, and identify each according to the name given in my listing.

Groups

All groups of order at most 32 are listed in

Here I reproduce all groups of order at most 32, numbered according to [TW].

nGroup Name in [TW] Notation in [TW] Name in my listing
  1   1/1 C1 (cyclic) C1
  2   2/1 C2 (cyclic) C2
  3   3/1 C3 (cyclic) C3
  4   4/1 C4 (cyclic) C4
  4   4/2 C22 (elementary abelian) C22
  5   5/1 C5 (cyclic) C5
  6   6/1 C6 (cyclic) C6
  6   6/2 D3 (dihedral) D3
  7   7/1 C7 (cyclic) C7
  8   8/1 C8 (cyclic) 8.1.8.0
  8   8/2 C2 × C4 8.3.8.0
  8   8/3 C23 (elementary abelian) 8.7.8.0
  8   8/4 D4 (dihedral) 8.5.2.0
  8   8/5 Q8 (quaternion) 8.1.2.0
  9   9/1 C9 (cyclic) C9
  9   9/2 C32 (elementary abelian) C32
  10   10/1 C10 (cyclic) C10
  10   10/2 D5 (dihedral) D5
  11   11/1 C11 (cyclic) C11
  12   12/1 C12 (cyclic) 12.1.12.0
  12   12/2 C2 × C6 12.3.12.0
  12   12/3 D6 (dihedral) 12.7.2.0
  12   12/4 A4 (alternating) 12.3.1.0
  12   12/5 Q6 (dicyclic) 12.1.2.0
  13   13/1 C13 (cyclic) C13
  14   14/1 C14 (cyclic) C14
  14   14/2 D7 (dihedral) D7
  15   15/1 C15 (cyclic) 15.2.15.0
  16   16/1 C16 (cyclic) 16.1.16.0
  16   16/2 C2 × C8 16.3.16.0
  16   16/3 C42 (homocyclic) 16.3.16.1
  16   16/4 C22 × C4 16.7.16.0
  16   16/5 C24 (elementary abelian) 16.15.16.0
  16   16/6 D4 × C2 = Γ2a1 16.11.4.6
  16   16/7 Q × C2 = Γ2a2 16.3.4.21
  16   16/8 Γ2b 16.7.4.74
  16   16/9 Γ2c1 16.7.4.21
  16   16/10 Γ2c2 16.3.4.22
  16   16/11 Γ2d 16.3.4.23
  16   16/12 D8 (dihedral) = Γ3a1 16.9.2.8
  16   16/13 Γ3a2 16.5.2.17
  16   16/14 Q8 (dicyclic) = Γ3a3 16.1.2.8
  17   17/1 C17 (cyclic) C17
  18   18/1 C18 (cyclic) 18.1.18.0
  18   18/2 C3 × C6 18.1.18.1
  18   18/3 S3 × C3 18.3.3.1
  18   18/4 D9 (dihedral) 18.9.1.1
  18   18/5 C32 : C2 18.9.1.0
  19   19/1 C19 (cyclic) C19
  20   20/1 C20 (cyclic) 20.1.20.0
  20   20/2 C2 × C10 20.3.20.0
  20   20/3 D10 (dihedral) 20.11.2.0
  20   20/4 Q10 (dicyclic) 20.1.2.0
  20   20/5 Hol(C5) 20.5.1.0
  21   21/1 C21 (cyclic) 21.2.21.0
  21   21/2 C7 : C3 21.14.1.2
  22   22/1 C22 (cyclic) C22
  22   22/2 D11 (dihedral) D11
  23   23/1 C23 (cyclic) C23
  24   24/1 C24 (cyclic) 24.1.24.0
  24   24/2 C2 × C12 24.3.24.0
  24   24/3 C22 × C6 24.7.24.0
  24   24/4 C2 × D6 24.15.4.0
  24   24/5 C2 × A4 24.7.2.8
  24   24/6 C2 × Q6 24.3.4.0
  24   24/7 C3 × D4 24.5.6.0
  24   24/8 C3 × Q8 24.1.6.0
  24   24/9 C4 × S3 24.7.4.0
  24   24/10 D12 (dihedral) 24.13.2.4
  24   24/11 Q12 (dicyclic) 24.1.2.5
  24   24/12 S4 (symmetric) 24.9.1.1
  24   24/13 SL2(3) 24.1.2.7
  24   24/14 C3 : C8 (semidirect) 24.1.4.2
  24   24/15 C3 : D4 (semidirect) 24.9.2.3
  25   25/1 C25 (cyclic) C25
  25   25/2 C52 (elementary abelian) C52
  26   26/1 C26 (cyclic) C26
  26   26/2 D13 (dihedral) D13
  27   27/1 C27 (cyclic) 27.2.27.0
  27   27/2 C3 × C9 27.8.27.0
  27   27/3 C33 (elementary abelian) 27.26.27.0
  27   27/4 3−1+2 27.26.3.0
  27   27/5 3+1+2 27.8.3.2
  28   28/1 C28 (cyclic) 28.1.28.0
  28   28/2 C2 × C14 28.3.28.0
  28   28/3 D14 (dihedral) 28.15.2.0
  28   28/4 Q14 (dicyclic) 28.1.2.0
  29   29/1 C29 (cyclic) C29
  30   30/1 C30 (cyclic) 30.1.30.0
  30   30/2 C3 × D5 30.5.3.0
  30   30/3 C5 × D3 30.3.5.0
  30   30/4 D15 (dihedral) 30.15.1.0
  31   31/1 C31 (cyclic) C31
  32   32/1 C32 (cyclic) C32
  32   32/2 C2 × C16 C2 × C16
  32   32/3 C4 × C8 C4 × C8
  32   32/4 C2 × C2 × C8 C2 × C2 × C8
  32   32/5 C2 × C4 × C4 C2 × C4 × C4
  32   32/6 C2 × C2 × C2 × C4 C2 × C2 × C2 × C4
  32   32/7 C25 (elementary abelian) C25
  32   32/8 D4 × C2 × C2,  Γ2a1 D4 × C2 × C2
  32   32/9 Q × C2 × C2,  Γ2a2 Q × C2 × C2
  32   32/10 Γ2b Γ2b
  32   32/11 Γ2c1 Γ2c1
  32   32/12 Γ2c2 Γ2c2
  32   32/13 Γ2d Γ2d
  32   32/14 D4 × C4,  Γ2e1 D4 × C4
  32   32/15 Q × C4,  Γ2e2 Q × C4
  32   32/16 Γ2f Γ2f
  32   32/17 Γ2g Γ2g
  32   32/18 Γ2h Γ2h
  32   32/19 Γ2i Γ2i
  32   32/20 Γ2j1 Γ2j1
  32   32/21 Γ2j2 Γ2j2
  32   32/22 Γ2k Γ2k
  32   32/23 D8 × C2,  Γ3a1 D8 × C2
  32   32/24 Γ3a2 Γ3a2
  32   32/25 Q8 × C2,  Γ3a3 Q8 × C2
  32   32/26 Γ3b Γ3b
  32   32/27 Γ3c1 Γ3c1
  32   32/28 Γ3c2 Γ3c2
  32   32/29 Γ3d1 Γ3d1
  32   32/30 Γ3d2 Γ3d2
  32   32/31 Γ3e Γ3e
  32   32/32 Γ3f Γ3f
  32   32/33 Γ4a1 Γ4a1
  32   32/34 Γ4a2,  Dih(C4 × C4) Γ4a2
  32   32/35 Γ4a3 Γ4a3
  32   32/36 Γ4b1 Γ4b1
  32   32/37 Γ4b2 Γ4b2
  32   32/38 Γ4c1 Γ4c1
  32   32/39 Γ4c2 Γ4c2
  32   32/40 Γ4c3 Γ4c3
  32   32/41 Γ4d Γ4d
  32   32/42 Γ5a1 Γ5a1
  32   32/43 Γ5a2 Γ5a2
  32   32/44 Γ6a1 Γ6a1
  32   32/45 Γ6a2 Γ6a2
  32   32/46 Γ7a1 Γ7a1
  32   32/47 Γ7a2 Γ7a2
  32   32/48 Γ7a3 Γ7a3
  32   32/49 D16 (dihedral),  Γ8a1 D16
  32   32/50 Γ8a2 Γ8a2
  32   32/51 Q16 (dicyclic),  Γ8a3 Q16

Non-Associative Moufang Loops

All non-associative Moufang loops of order less than 64 are listed in

nLoop Name in [GMR] Notation in [GMR] Name in my listing
  12   12/1 M12(S3,2) 12.9.1.1
  16   16/1 M16(D4,2) 16.13.2.6
  16   16/2 M16(Q,2) 16.9.2.435
  16   16/3 M16(Q) 16.1.2.31
  16   16/4 M16(C2 × C4) 16.9.2.13
  16   16/5 M16(C2 × C4,Q) 16.5.2.279
  20   20/1 M20(D5,2) 20.15.1.1
  24   24/1 M24(D6,2) 24.19.2.2
  24   24/2 M24(A4,2) 24.15.1.0
  24   24/3 M24(Q6,2) 24.13.2.6
  24   24/4 M24(G12,C2 × C4) 24.7.2.11
  24   24/5 M24(G12,Q8) 24.1.2.9
  28   28/1 M28(D7,2) 28.21.1.0

Non-Moufang Bol Loops

A list of Non-Moufang Bol loops of order less than 32 (complete for orders n not equal to 16, 24, 27, 30) appears in

nLoop Name in [GM] Notation in [GM] Name in my listing
  8   8/1 B8(Π1) 8.1.4.0
  8   8/2 B8(Π2) 8.3.2.0
  8   8/3 B8(Π3) 8.3.2.1
  8   8/4 B8(Π4) 8.5.4.1
  8   8/5 B8(Π5) 8.5.4.0
  8   8/6 B8(Π6) 8.7.2.0
  12   12/1 B12(Π,3) 12.5.3.0
  12   12/2 B12(αβΠ,3) 12.9.1.0
  15   15/1 NR15(3,5,1,2) 15.10.1.0
  15   15/2 NR15(3,5,3,3) 15.10.1.1
  16   16/1 B8(Π1) × C2 16.3.8.0
  16   16/2 B8(Π2) × C2 16.7.4.0
  16   16/3 B8(Π3) × C2 16.7.4.10
  16   16/4 B8(Π4) × C2 16.11.8.0
  16   16/5 B8(Π5) × C2 16.11.8.1
  16   16/6 B8(Π6) × C2 16.15.4.0
  16   16/7 B16(Π,4) 16.7.4.16
  16   16/8 B16(αβΠ,4) 16.13.2.2
  16   16/9 B16(αΓΠ,4) 16.5.2.0
  16   16/10 B16(22,2,4) 16.9.6.0
  16   16/11 B16(22,4,4) 16.13.2.81
  16   16/12 C2 ×θ B8(Π6) 16.11.2.146
  18   18/1 B18 18.3.3.0
  20   20/1 B20(Π,5) 20.7.3.0
  20   20/2 B20(αβΠ,5) 20.15.1.0
  21   21/1 NR21(3,7,1,3) 21.14.1.0
  21   21/2 NR21(3,7,5,5) 21.14.1.1
  24   24/1 B8(Π1) × C3 24.1.12.0
  24   24/2 B8(Π2) × C3 24.3.6.1
  24   24/3 B8(Π3) × C3 24.3.6.2
  24   24/4 B8(Π4) × C3 24.5.12.1
  24   24/5 B8(Π5) × C3 24.5.12.0
  24   24/6 B8(Π6) × C3 24.7.6.1
  24   24/7 B24(Π,6) 24.9.4.0
  24   24/8 B24(αΓΠ,6) 24.7.2.0
  24   24/9 B24(23,2,3) 24.9.7.0
  24   24/10 B24(23,3,3) 24.11.6.0
  24   24/11 B24(23,4,3) 24.13.5.0
  24   24/12 B24(23,5,3) 24.15.4.1
  24   24/13 B24(23,6,3) 24.17.3.0
  24   24/14 B24(23,7,3) 24.19.2.0
  24   24/15 B24(23,8,3) 24.21.1.0
  24   24/16 B8(Π6) ×θ1 C3 24.9.1.0
  24   24/17 B8(Π6) ×θ2 C3 24.11.2.0
  24   24/18 B8(Π6) ×θ3 C3 24.13.1.0
  24   24/19 B8(Π6) ×θ5 C3 24.17.1.0
  24   24/20 B8(Π6) ×θ6 C3 24.19.2.1
  24   24/21 B8(Π6) ×θ7 C3 24.21.1.1
  27   27/1 NR27(3,9,1,4) 27.20.3.0
  27   27/2 NR27(3,9,1,7) 27.8.3.0
  27   27/3 NR27(3,9,4,4) 27.8.3.1
  27   27/4 NR27(3,9,7,7) 27.20.3.1
  28   28/1 B28(Π,7) 28.9.3.0
  28   28/2 B28(αβΠ,7) 28.21.1.1
  30   30/1 NR15(3,5,1,2) × C2 30.1.2.0
  30   30/2 NR15(3,5,3,3) × C2 30.1.2.1

/ revised February, 2005