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.
Here I reproduce all groups of order at most 32, numbered according to [TW].
n | Group Name in [TW] | Notation in [TW] | Name in my listing |
---|---|---|---|
1 | 1/1 | C_{1} (cyclic) | C_{1} |
2 | 2/1 | C_{2} (cyclic) | C_{2} |
3 | 3/1 | C_{3} (cyclic) | C_{3} |
4 | 4/1 | C_{4} (cyclic) | C_{4} |
4 | 4/2 | C_{2}^{2} (elementary abelian) | C_{2}^{2} |
5 | 5/1 | C_{5} (cyclic) | C_{5} |
6 | 6/1 | C_{6} (cyclic) | C_{6} |
6 | 6/2 | D_{3} (dihedral) | D_{3} |
7 | 7/1 | C_{7} (cyclic) | C_{7} |
8 | 8/1 | C_{8} (cyclic) | 8.1.8.0 |
8 | 8/2 | C_{2} × C_{4} | 8.3.8.0 |
8 | 8/3 | C_{2}^{3} (elementary abelian) | 8.7.8.0 |
8 | 8/4 | D_{4} (dihedral) | 8.5.2.0 |
8 | 8/5 | Q_{8} (quaternion) | 8.1.2.0 |
9 | 9/1 | C_{9} (cyclic) | C_{9} |
9 | 9/2 | C_{3}^{2} (elementary abelian) | C_{3}^{2} |
10 | 10/1 | C_{10} (cyclic) | C_{10} |
10 | 10/2 | D_{5} (dihedral) | D_{5} |
11 | 11/1 | C_{11} (cyclic) | C_{11} |
12 | 12/1 | C_{12} (cyclic) | 12.1.12.0 |
12 | 12/2 | C_{2} × C_{6} | 12.3.12.0 |
12 | 12/3 | D_{6} (dihedral) | 12.7.2.0 |
12 | 12/4 | A_{4} (alternating) | 12.3.1.0 |
12 | 12/5 | Q_{6} (dicyclic) | 12.1.2.0 |
13 | 13/1 | C_{13} (cyclic) | C_{13} |
14 | 14/1 | C_{14} (cyclic) | C_{14} |
14 | 14/2 | D_{7} (dihedral) | D_{7} |
15 | 15/1 | C_{15} (cyclic) | 15.2.15.0 |
16 | 16/1 | C_{16} (cyclic) | 16.1.16.0 |
16 | 16/2 | C_{2} × C_{8} | 16.3.16.0 |
16 | 16/3 | C_{4}^{2} (homocyclic) | 16.3.16.1 |
16 | 16/4 | C_{2}^{2} × C_{4} | 16.7.16.0 |
16 | 16/5 | C_{2}^{4} (elementary abelian) | 16.15.16.0 |
16 | 16/6 | D_{4} × C_{2} = Γ_{2}a_{1} | 16.11.4.6 |
16 | 16/7 | Q × C_{2} = Γ_{2}a_{2} | 16.3.4.21 |
16 | 16/8 | Γ_{2}b | 16.7.4.74 |
16 | 16/9 | Γ_{2}c_{1} | 16.7.4.21 |
16 | 16/10 | Γ_{2}c_{2} | 16.3.4.22 |
16 | 16/11 | Γ_{2}d | 16.3.4.23 |
16 | 16/12 | D_{8} (dihedral) = Γ_{3}a_{1} | 16.9.2.8 |
16 | 16/13 | Γ_{3}a_{2} | 16.5.2.17 |
16 | 16/14 | Q_{8} (dicyclic) = Γ_{3}a_{3} | 16.1.2.8 |
17 | 17/1 | C_{17} (cyclic) | C_{17} |
18 | 18/1 | C_{18} (cyclic) | 18.1.18.0 |
18 | 18/2 | C_{3} × C_{6} | 18.1.18.1 |
18 | 18/3 | S_{3} × C_{3} | 18.3.3.1 |
18 | 18/4 | D_{9} (dihedral) | 18.9.1.1 |
18 | 18/5 | C_{3}^{2} : C_{2} | 18.9.1.0 |
19 | 19/1 | C_{19} (cyclic) | C_{19} |
20 | 20/1 | C_{20} (cyclic) | 20.1.20.0 |
20 | 20/2 | C_{2} × C_{10} | 20.3.20.0 |
20 | 20/3 | D_{10} (dihedral) | 20.11.2.0 |
20 | 20/4 | Q_{10} (dicyclic) | 20.1.2.0 |
20 | 20/5 | Hol(C_{5}) | 20.5.1.0 |
21 | 21/1 | C_{21} (cyclic) | 21.2.21.0 |
21 | 21/2 | C_{7} : C_{3} | 21.14.1.2 |
22 | 22/1 | C_{22} (cyclic) | C_{22} |
22 | 22/2 | D_{11} (dihedral) | D_{11} |
23 | 23/1 | C_{23} (cyclic) | C_{23} |
24 | 24/1 | C_{24} (cyclic) | 24.1.24.0 |
24 | 24/2 | C_{2} × C_{12} | 24.3.24.0 |
24 | 24/3 | C_{2}^{2} × C_{6} | 24.7.24.0 |
24 | 24/4 | C_{2} × D_{6} | 24.15.4.0 |
24 | 24/5 | C_{2} × A_{4} | 24.7.2.8 |
24 | 24/6 | C_{2} × Q_{6} | 24.3.4.0 |
24 | 24/7 | C_{3} × D_{4} | 24.5.6.0 |
24 | 24/8 | C_{3} × Q_{8} | 24.1.6.0 |
24 | 24/9 | C_{4} × S_{3} | 24.7.4.0 |
24 | 24/10 | D_{12} (dihedral) | 24.13.2.4 |
24 | 24/11 | Q_{12} (dicyclic) | 24.1.2.5 |
24 | 24/12 | S_{4} (symmetric) | 24.9.1.1 |
24 | 24/13 | SL_{2}(3) | 24.1.2.7 |
24 | 24/14 | C_{3} : C_{8} (semidirect) | 24.1.4.2 |
24 | 24/15 | C_{3} : D_{4} (semidirect) | 24.9.2.3 |
25 | 25/1 | C_{25} (cyclic) | C_{25} |
25 | 25/2 | C_{5}^{2} (elementary abelian) | C_{5}^{2} |
26 | 26/1 | C_{26} (cyclic) | C_{26} |
26 | 26/2 | D_{13} (dihedral) | D_{13} |
27 | 27/1 | C_{27} (cyclic) | 27.2.27.0 |
27 | 27/2 | C_{3} × C_{9} | 27.8.27.0 |
27 | 27/3 | C_{3}^{3} (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 | C_{28} (cyclic) | 28.1.28.0 |
28 | 28/2 | C_{2} × C_{14} | 28.3.28.0 |
28 | 28/3 | D_{14} (dihedral) | 28.15.2.0 |
28 | 28/4 | Q_{14} (dicyclic) | 28.1.2.0 |
29 | 29/1 | C_{29} (cyclic) | C_{29} |
30 | 30/1 | C_{30} (cyclic) | 30.1.30.0 |
30 | 30/2 | C_{3} × D_{5} | 30.5.3.0 |
30 | 30/3 | C_{5} × D_{3} | 30.3.5.0 |
30 | 30/4 | D_{15} (dihedral) | 30.15.1.0 |
31 | 31/1 | C_{31} (cyclic) | C_{31} |
32 | 32/1 | C_{32} (cyclic) | C_{32} |
32 | 32/2 | C_{2} × C_{16} | C_{2} × C_{16} |
32 | 32/3 | C_{4} × C_{8} | C_{4} × C_{8} |
32 | 32/4 | C_{2} × C_{2} × C_{8} | C_{2} × C_{2} × C_{8} |
32 | 32/5 | C_{2} × C_{4} × C_{4} | C_{2} × C_{4} × C_{4} |
32 | 32/6 | C_{2} × C_{2} × C_{2} × C_{4} | C_{2} × C_{2} × C_{2} × C_{4} |
32 | 32/7 | C_{2}^{5} (elementary abelian) | C_{2}^{5} |
32 | 32/8 | D_{4} × C_{2} × C_{2}, Γ_{2}a_{1} | D_{4} × C_{2} × C_{2} |
32 | 32/9 | Q × C_{2} × C_{2}, Γ_{2}a_{2} | Q × C_{2} × C_{2} |
32 | 32/10 | Γ_{2}b | Γ_{2}b |
32 | 32/11 | Γ_{2}c_{1} | Γ_{2}c_{1} |
32 | 32/12 | Γ_{2}c_{2} | Γ_{2}c_{2} |
32 | 32/13 | Γ_{2}d | Γ_{2}d |
32 | 32/14 | D_{4} × C_{4}, Γ_{2}e_{1} | D_{4} × C_{4} |
32 | 32/15 | Q × C_{4}, Γ_{2}e_{2} | Q × C_{4} |
32 | 32/16 | Γ_{2}f | Γ_{2}f |
32 | 32/17 | Γ_{2}g | Γ_{2}g |
32 | 32/18 | Γ_{2}h | Γ_{2}h |
32 | 32/19 | Γ_{2}i | Γ_{2}i |
32 | 32/20 | Γ_{2}j_{1} | Γ_{2}j_{1} |
32 | 32/21 | Γ_{2}j_{2} | Γ_{2}j_{2} |
32 | 32/22 | Γ_{2}k | Γ_{2}k |
32 | 32/23 | D_{8} × C_{2}, Γ_{3}a_{1} | D_{8} × C_{2} |
32 | 32/24 | Γ_{3}a_{2} | Γ_{3}a_{2} |
32 | 32/25 | Q_{8} × C_{2}, Γ_{3}a_{3} | Q_{8} × C_{2} |
32 | 32/26 | Γ_{3}b | Γ_{3}b |
32 | 32/27 | Γ_{3}c_{1} | Γ_{3}c_{1} |
32 | 32/28 | Γ_{3}c_{2} | Γ_{3}c_{2} |
32 | 32/29 | Γ_{3}d_{1} | Γ_{3}d_{1} |
32 | 32/30 | Γ_{3}d_{2} | Γ_{3}d_{2} |
32 | 32/31 | Γ_{3}e | Γ_{3}e |
32 | 32/32 | Γ_{3}f | Γ_{3}f |
32 | 32/33 | Γ_{4}a_{1} | Γ_{4}a_{1} |
32 | 32/34 | Γ_{4}a_{2}, Dih(C_{4} × C_{4}) | Γ_{4}a_{2} |
32 | 32/35 | Γ_{4}a_{3} | Γ_{4}a_{3} |
32 | 32/36 | Γ_{4}b_{1} | Γ_{4}b_{1} |
32 | 32/37 | Γ_{4}b_{2} | Γ_{4}b_{2} |
32 | 32/38 | Γ_{4}c_{1} | Γ_{4}c_{1} |
32 | 32/39 | Γ_{4}c_{2} | Γ_{4}c_{2} |
32 | 32/40 | Γ_{4}c_{3} | Γ_{4}c_{3} |
32 | 32/41 | Γ_{4}d | Γ_{4}d |
32 | 32/42 | Γ_{5}a_{1} | Γ_{5}a_{1} |
32 | 32/43 | Γ_{5}a_{2} | Γ_{5}a_{2} |
32 | 32/44 | Γ_{6}a_{1} | Γ_{6}a_{1} |
32 | 32/45 | Γ_{6}a_{2} | Γ_{6}a_{2} |
32 | 32/46 | Γ_{7}a_{1} | Γ_{7}a_{1} |
32 | 32/47 | Γ_{7}a_{2} | Γ_{7}a_{2} |
32 | 32/48 | Γ_{7}a_{3} | Γ_{7}a_{3} |
32 | 32/49 | D_{16} (dihedral), Γ_{8}a_{1} | D_{16} |
32 | 32/50 | Γ_{8}a_{2} | Γ_{8}a_{2} |
32 | 32/51 | Q_{16} (dicyclic), Γ_{8}a_{3} | Q_{16} |
n | Loop Name in [GMR] | Notation in [GMR] | Name in my listing |
---|---|---|---|
12 | 12/1 | M_{12}(S_{3},2) | 12.9.1.1 |
16 | 16/1 | M_{16}(D_{4},2) | 16.13.2.6 |
16 | 16/2 | M_{16}(Q,2) | 16.9.2.435 |
16 | 16/3 | M_{16}(Q) | 16.1.2.31 |
16 | 16/4 | M_{16}(C_{2} × C_{4}) | 16.9.2.13 |
16 | 16/5 | M_{16}(C_{2} × C_{4},Q) | 16.5.2.279 |
20 | 20/1 | M_{20}(D_{5},2) | 20.15.1.1 |
24 | 24/1 | M_{24}(D_{6},2) | 24.19.2.2 |
24 | 24/2 | M_{24}(A_{4},2) | 24.15.1.0 |
24 | 24/3 | M_{24}(Q_{6},2) | 24.13.2.6 |
24 | 24/4 | M_{24}(G_{12},C_{2} × C_{4}) | 24.7.2.11 |
24 | 24/5 | M_{24}(G_{12},Q_{8}) | 24.1.2.9 |
28 | 28/1 | M_{28}(D_{7},2) | 28.21.1.0 |
n | Loop Name in [GM] | Notation in [GM] | Name in my listing |
---|---|---|---|
8 | 8/1 | B_{8}(Π_{1}) | 8.1.4.0 |
8 | 8/2 | B_{8}(Π_{2}) | 8.3.2.0 |
8 | 8/3 | B_{8}(Π_{3}) | 8.3.2.1 |
8 | 8/4 | B_{8}(Π_{4}) | 8.5.4.1 |
8 | 8/5 | B_{8}(Π_{5}) | 8.5.4.0 |
8 | 8/6 | B_{8}(Π_{6}) | 8.7.2.0 |
12 | 12/1 | B_{12}(Π,3) | 12.5.3.0 |
12 | 12/2 | B_{12}(αβΠ,3) | 12.9.1.0 |
15 | 15/1 | NR_{15}(3,5,1,2) | 15.10.1.0 |
15 | 15/2 | NR_{15}(3,5,3,3) | 15.10.1.1 |
16 | 16/1 | B_{8}(Π_{1}) × C_{2} | 16.3.8.0 |
16 | 16/2 | B_{8}(Π_{2}) × C_{2} | 16.7.4.0 |
16 | 16/3 | B_{8}(Π_{3}) × C_{2} | 16.7.4.10 |
16 | 16/4 | B_{8}(Π_{4}) × C_{2} | 16.11.8.0 |
16 | 16/5 | B_{8}(Π_{5}) × C_{2} | 16.11.8.1 |
16 | 16/6 | B_{8}(Π_{6}) × C_{2} | 16.15.4.0 |
16 | 16/7 | B_{16}(Π,4) | 16.7.4.16 |
16 | 16/8 | B_{16}(αβΠ,4) | 16.13.2.2 |
16 | 16/9 | B_{16}(αΓΠ,4) | 16.5.2.0 |
16 | 16/10 | B_{16}(2^{2},2,4) | 16.9.6.0 |
16 | 16/11 | B_{16}(2^{2},4,4) | 16.13.2.81 |
16 | 16/12 | C_{2} ×_{θ} B_{8}(Π_{6}) | 16.11.2.146 |
18 | 18/1 | B_{18} | 18.3.3.0 |
20 | 20/1 | B_{20}(Π,5) | 20.7.3.0 |
20 | 20/2 | B_{20}(αβΠ,5) | 20.15.1.0 |
21 | 21/1 | NR_{21}(3,7,1,3) | 21.14.1.0 |
21 | 21/2 | NR_{21}(3,7,5,5) | 21.14.1.1 |
24 | 24/1 | B_{8}(Π_{1}) × C_{3} | 24.1.12.0 |
24 | 24/2 | B_{8}(Π_{2}) × C_{3} | 24.3.6.1 |
24 | 24/3 | B_{8}(Π_{3}) × C_{3} | 24.3.6.2 |
24 | 24/4 | B_{8}(Π_{4}) × C_{3} | 24.5.12.1 |
24 | 24/5 | B_{8}(Π_{5}) × C_{3} | 24.5.12.0 |
24 | 24/6 | B_{8}(Π_{6}) × C_{3} | 24.7.6.1 |
24 | 24/7 | B_{24}(Π,6) | 24.9.4.0 |
24 | 24/8 | B_{24}(αΓΠ,6) | 24.7.2.0 |
24 | 24/9 | B_{24}(2^{3},2,3) | 24.9.7.0 |
24 | 24/10 | B_{24}(2^{3},3,3) | 24.11.6.0 |
24 | 24/11 | B_{24}(2^{3},4,3) | 24.13.5.0 |
24 | 24/12 | B_{24}(2^{3},5,3) | 24.15.4.1 |
24 | 24/13 | B_{24}(2^{3},6,3) | 24.17.3.0 |
24 | 24/14 | B_{24}(2^{3},7,3) | 24.19.2.0 |
24 | 24/15 | B_{24}(2^{3},8,3) | 24.21.1.0 |
24 | 24/16 | B_{8}(Π_{6}) ×_{θ1} C_{3} | 24.9.1.0 |
24 | 24/17 | B_{8}(Π_{6}) ×_{θ2} C_{3} | 24.11.2.0 |
24 | 24/18 | B_{8}(Π_{6}) ×_{θ3} C_{3} | 24.13.1.0 |
24 | 24/19 | B_{8}(Π_{6}) ×_{θ5} C_{3} | 24.17.1.0 |
24 | 24/20 | B_{8}(Π_{6}) ×_{θ6} C_{3} | 24.19.2.1 |
24 | 24/21 | B_{8}(Π_{6}) ×_{θ7} C_{3} | 24.21.1.1 |
27 | 27/1 | NR_{27}(3,9,1,4) | 27.20.3.0 |
27 | 27/2 | NR_{27}(3,9,1,7) | 27.8.3.0 |
27 | 27/3 | NR_{27}(3,9,4,4) | 27.8.3.1 |
27 | 27/4 | NR_{27}(3,9,7,7) | 27.20.3.1 |
28 | 28/1 | B_{28}(Π,7) | 28.9.3.0 |
28 | 28/2 | B_{28}(αβΠ,7) | 28.21.1.1 |
30 | 30/1 | NR_{15}(3,5,1,2) × C_{2} | 30.1.2.0 |
30 | 30/2 | NR_{15}(3,5,3,3) × C_{2} | 30.1.2.1 |