  Primary Cyclic matrices were used (but not named) by Holt and Rees in their
version of Parker's MEAT-AXE algorithm to test irreducibility of finite matrix
groups and algebras. They are matrices $X$ with at least one cyclic component
in the primary decomposition of the underlying vector space as an $X$-module.
Let $\operatorname{M}(c,q^b)$ be an irreducible subalgebra of
$\operatorname{M}(n,q)$, where $n=bc >c$. We prove a generalisation of the
Kung-Stong Cycle Index, and use it to obtain a lower bound for the proportion
of primary cyclic matrices in $\operatorname{M}(c,q^b)$. This extends work of
Glasby and the second author on the case $b=1$.
