Non-power-associative loop of order five

This article is about a particular loop, viz a loop unique upto isomorphism
View a complete list of particular loops

Definition

This is a loop with five elements. Up to isomorphism, it is the only loop of five elements that is not a power-associative loop.

$*$	1	2	3	4	5
1	1	2	3	4	5
2	2	1	5	3	4
3	3	5	4	2	1
4	4	3	1	5	2
5	5	4	2	1	3

In other words, this is the algebra loop corresponding to the Latin square:

${\begin{pmatrix}1&2&3&4&5\\2&1&5&3&4\\3&5&4&2&1\\4&3&1&5&2\\5&4&2&1&3\\\end{pmatrix}}$

Algebra loop properties

Property	Satisfied?	Explanation
power-associative loop	No	$\!3*(3*3)\neq (3*3)*3$
flexible loop	No	$\!3*(3*3)\neq (3*3)*3$
alternative loop	No	$\!3*(3*3)\neq (3*3)*3$
Moufang loop	No	(since it's not alternative)
left Bol loop	No	(since it's not flexible)
monogenic loop	Yes	generated by one element

Arithmetic functions

Function	Value	Explanation
order	5