A Moufang loop $M$ is a loop that satisfies the Moufang identity:
$(zx)(yz) = z((xy)z),\forall x,y,z\in M$.
From here, we get the following statement:
For any field $F$ let $M(F)$ denote the Moufang loop of unit norm elements in the (unique) split-octonion algebra over $F$. Let $Z$ denote the center of $M(F)$. If the characteristic of $F$ is $2$ then $Z = \{e\}$, otherwise $Z = \{±e\}$. The Paige loop over $F$ is the loop $M^*(F) = M(F)/Z$. Paige loops are nonassociative simple Moufang loops. All finite nonassociative simple Moufang loops are Paige loops over finite fields.
So we have (see multiplication table below for $i,\ell, k$), $M(\mathbb{F}_{p^n})=\{x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell +x_{5}\,\ell i+x_{6}\,\ell j+x_{7}\,\ell k\}$, where $x_m\in \mathbb{F}_{p^n}$ and unit norm:
$N(x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell +x_{5}\,\ell i+x_{6}\,\ell j+x_{7}\,\ell k)=(x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell +x_{5}\,\ell i+x_{6}\,\ell j+x_{7}\,\ell k)\cdot (x_{0}-x_{1}\,i-x_{2}\,j-x_{3}\,k-x_{4}\,\ell -x_{5}\,\ell i-x_{6}\,\ell j-x_{7}\,\ell k)=1$
Question: What are the orders of these Paige loops?
Examples:
$|M^*(\mathbb{F}_2)|=120$
$|M^*(\mathbb{F}_3)|=624/2=312$
It gets much harder to manually compute; is there work on this?
From page 12 of this, we have that for $q=p^k$
$|M^*(\mathbb{F}_{q})|=\frac{q^3(q^4-1)}{\gcd (q+1,2)}$.