Bol Loops of Order 21

I've been working on a classification of Bol loops of small order. Here is a list of the 4 Bol loops of order 21. This includes the two groups of order 21 and the 2 non-associative Bol loops of order 21 (neither of which is Moufang). The two non-associative examples are isotopic to one another. These examples have been known for a long time, and in 2002 I performed a simple backtrack enumeration to check that this list is complete.

I have made available

a quick review of the relevant definitions;
a suitable list of references for further explanation, proofs, etc.;
a cross-reference list comparing this list with previously published lists of groups and loops;
descriptions of these loops in html format, found through the links below; and
the Cayley tables (5 KB) in plain text form.

In listing elements of the commutator (resp. associator) subloop of each of these loops, we have printed in italics any elements which are not actual commutators (resp. associators). (I haven't checked, however, whether in fact this phenomenon occurs among any the loops of order 21 in our list.)

The 3 Isotopy Classes of Bol Loops

Isotopy Class 0: The Cyclic Group 21.2.21.0

Isotopy Class 1: The Frobenius Group 21.14.1.2

Isotopy Class 2: The 2 Non-Moufang (non-associative) Bol Loops 21.14.1.0, 21.14.1.1

Naming of the Loops

For each of the loops of order 21, I have used a name 21.i.c.k where i is the number of elements of order 3, c=|C(L)| and the index k=0,1,2,... indicates merely the order in which each isomorphism class of loop was first encountered by my computer.

/ revised February, 2005