In the example below a 2x2 Latin square is multiplied by a 3x3 Latin square which gives a 6x6 Latin square. My question is what it the method for multiplying these two different sized Latin squares together, how do you arrive at the 6x6 Latin Square and where does each number come from?
1 2
2 1
2x2 Latin square
X
3x3 Latin square
1 2 3
2 3 1
3 1 2
=
1 2 3 4 5 6
2 1 4 3 6 5
3 4 5 6 1 2
4 3 6 5 2 1
5 6 1 2 3 4
6 5 2 1 4 3
As stated in the book, this is the result of identifying Latin squares with quasigroups; this operation then corresponds to the direct product of quasigroups. In this example, let's say we have quasigroups $G = \{g_1,g_2\}$ and $H = \{h_1,h_2,h_3\}$ with multiplication tables given by those two matrices. Then the direct product $$G \times H = \{ (g_1,h_1),(g_2,h_1),(g_1,h_2), (g_2,h_2), (g_1,h_3), (g_2,h_3)\}$$ may be identified with the set $\{1,2,3,4,5,6\}$ (this ordering is arbitrary but important; I'm using the one the author picked). Then for example, the element in the fourth line and third column in the product square is given by the product $(g_2,h_2)(g_1,h_2) = (g_2g_1,h_2h_2) = (g_2,h_3)$, which corresponds to the label $6$.