# Bol Loops of Order 30

I've been working on a classification of Bol loops of small order.
Here is a list of the 6 *known* Bol loops of order 30. This includes the 4 groups of order 30 and the
2 known non-associative Bol loops of order 30 (neither of which is Moufang). The two known non-associative
examples are isotopic to one another.

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 (15 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 30 in our list.)

**Isotopy Class 0: The 2 ***Known* Non-associative Bol Loops
30.1.2.0,
30.1.2.1

**Isotopy Classes 1,2,3,4: The 4 Groups**
30.1.30.0
30.3.5.0
30.5.3.0
30.15.1.0

### Naming of the Loops

For each of the loops of order 30, I have used a name 30.*i*.*c*.*k* where *i*=|*I*(*L*)|,
*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 March, 2004