Sandpile groups of Cayley graphs of 𝔽^r₂

Figures & data

Fig. 1 Examples of Cayley graphs in dimension r = 3.

Table 1 Sylow-2 cyclic factors of $K(Q_n)$ for $n\le 11$ (table from [2]).

n	${\text{Syl}}_2(K(Q_n))$
2	${\mathbb{Z}}_4$
3	${\mathbb{Z}}_2{\mathbb{Z}}_8^2$
4	${\mathbb{Z}}_2^2{\mathbb{Z}}_8^4{\mathbb{Z}}_{32}$
5	${\mathbb{Z}}_2^6{\mathbb{Z}}_8^4{\mathbb{Z}}_{16}{\mathbb{Z}}_{64}^4$
6	${\mathbb{Z}}_2^{12}{\mathbb{Z}}_4^4{\mathbb{Z}}_8{\mathbb{Z}}_{32}^4{\mathbb{Z}}_{64}^{10}$
7	${\mathbb{Z}}_2^{28}{\mathbb{Z}}_4{\mathbb{Z}}_{16}^8{\mathbb{Z}}_{32}^6{\mathbb{Z}}_{64}^{14}{\mathbb{Z}}_{128}^6$
8	${\mathbb{Z}}_2^{56}{\mathbb{Z}}_4^2{\mathbb{Z}}_{16}^{16}{\mathbb{Z}}_{32}^{12}{\mathbb{Z}}_{64}^{28}{\mathbb{Z}}_{128}^{12}{\mathbb{Z}}_{1024}$
9	${\mathbb{Z}}_2^{120}{\mathbb{Z}}_4^{10}{\mathbb{Z}}_{16}^{16}{\mathbb{Z}}_{32}^{26}{\mathbb{Z}}_{64}^{48}{\mathbb{Z}}_{128}^{26}{\mathbb{Z}}_{512}{\mathbb{Z}}_{2048}^8$
10	${\mathbb{Z}}_2^{240}{\mathbb{Z}}_4^{36}{\mathbb{Z}}_8^{26}{\mathbb{Z}}_{32}^{16}{\mathbb{Z}}_{64}^{148}{\mathbb{Z}}_{256}{\mathbb{Z}}_{1024}^{26}{\mathbb{Z}}_{2048}^{18}$
11	${\mathbb{Z}}_2^{496}{\mathbb{Z}}_4^{66}{\mathbb{Z}}_8^{32}{\mathbb{Z}}_{16}^{100}{\mathbb{Z}}_{64}^{164}{\mathbb{Z}}_{128}{\mathbb{Z}}_{512}^{100}{\mathbb{Z}}_{2048}^{64}$