# 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
• [TW] A.D. Thomas and G.V. Wood, Group Tables, Shiva Publishing, Kent, 1980.

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 31+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
• [GMR] Edgar G. Goodaire, Sean May and Maitreyi Raman, The Moufang Loops of Order less than 64, Nova Science, Commack, NY, 1999.

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
• [GM] Edgar G. Goodaire and Sean May, Bol Loops of Order less than 32, Dept. of Math and Statistics, Memorial University of Newfoundland, Canada, 1995.

nLoop Name in [GM] Notation in [GM] Name in my listing
8   8/1 B81) 8.1.4.0
8   8/2 B82) 8.3.2.0
8   8/3 B83) 8.3.2.1
8   8/4 B84) 8.5.4.1
8   8/5 B85) 8.5.4.0
8   8/6 B86) 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 B81) × C2 16.3.8.0
16   16/2 B82) × C2 16.7.4.0
16   16/3 B83) × C2 16.7.4.10
16   16/4 B84) × C2 16.11.8.0
16   16/5 B85) × C2 16.11.8.1
16   16/6 B86) × 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 B86) 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 B81) × C3 24.1.12.0
24   24/2 B82) × C3 24.3.6.1
24   24/3 B83) × C3 24.3.6.2
24   24/4 B84) × C3 24.5.12.1
24   24/5 B85) × C3 24.5.12.0
24   24/6 B86) × C3 24.7.6.1
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 B86) ×θ1 C3 24.9.1.0
24   24/17 B86) ×θ2 C3 24.11.2.0
24   24/18 B86) ×θ3 C3 24.13.1.0
24   24/19 B86) ×θ5 C3 24.17.1.0
24   24/20 B86) ×θ6 C3 24.19.2.1
24   24/21 B86) ×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