Over here it is claimed the group $G$ of permutations of a $4\times4$ grid generated by the subgroups $R$ of row permutations, $C$ of column permutations and $B$ of block permutations (of the four $2\times2$ quadrants) has size $|G|=322560=8\cdot8!$. Where does this number come from? Is there a nice standard way to represent its elements in terms of those from $R,C,B$?
I see $R,C,B\cong S_4$ and $RC$ is an internal direct product, and $RC\cap B$ is the Klein-four subgroup of $B\cong S_4$. However, the subset $(RC)B$ is not a subgroup, because its size is not a divisor - indeed $|G|/(|RC|\cdot|B|)=(2\cdot5\cdot7)/3$. (Note the size of $(RC)B$ would be the product $|RC||B|$ divided by $|RC\cap B|$.) Therefore, there are elements of $G$ which are more complicated than a straight product $rcb$ ($r\in R,c\in C,b\in B$). What's a good way to describe more elements of $G$?
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There is a narrower version of this that might shed light on the whole situation. I was thinking about the stabilizer subgroup $S$ of the diamond figure above. According the linked source, the orbit $GD$ has size $24\cdot35$ (i.e. that's how many figures emerge from applying elements of $G$ to the $D$), so then $|S|=|G|/|GD|=4!\cdot2^4$ by the orbit-stabilizer theorem. I would guess that $S\cong S_4\times\mathbb{Z}_2^4$, but I encounter a problem verifying this. The $B\subset S$ is the $S_4$, and there is a $H\cong\mathbb{Z}_2^4$ generated by swapping the first/third and second/fourth of the rows/columns. The problem is that $B\cap H$ is the Klein-four subgroup of $B\cong S_4$, so $S\ne B\times H$. What other elements of $S$ are there?
This itself can be narrowed. Consider the set $X$ of four tiles (in the diamond figure) in the upper left corner of each quadrant. These tiles are teal in the top left and black in the bottom right, and occur in odd-parity rows and columns. Then $S$ acts on $X$, and we have a quotient map $S\to S_X$. Its kernel $T$ must, in addition to stabilizing the diamond figure $D$, also fix $X$ pointwise. We ought to have $T\cong\mathbb{Z}_2^4$, but all I can see in $T$ is the swaps of even-parity rows/columns (which is only $\mathbb{Z}_2^2$).
I suppose an 'ultimate' answer might simply establish the stronger fact $G\cong\mathrm{Aff}_4\mathbb{F}_2$. I was hoping more elementary considerations for this simpler question could supplement my eventual understanding of this stronger fact, but that might not be realistic. At any rate, I don't really understand this isomorphism.
This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4x4 patterns.
What exactly is a "structure" here? It says there are $|GD|=24\times35$ diagrams but $35$ structures, so presumably $4!$ diagrams per structure. Are the "structures" the orbits of some copy of $S_4$ acting freely on $GD$? Also, when the author says the set of four tiles, do they mean the set of four types of tiles? (So, equivalently, the partition of the sixteen tiles into four groups of four.) And how does a line diagram "indicate" the "locations" of pairs of tile [types]?