Let $R$ be a ring (unital, commutative) and $M$ a free $R$-module of finite rank. A quadratic form is a map $q:M\rightarrow R$ such that
- $\forall r\in R:\forall m\in M: q(rm)=r^2\cdot q(m)$ and
- the induced map $\beta: M\times M\rightarrow R, (m_1,m_2)\mapsto q(m_1+m_2)-q(m_1)-q(m_2)$ is $R$-bilinear.
We say $(M,q)$ is a quadratic module (or a quadratic space, if $R$ is a field). We call $\beta$ the associated bilinear form.
One defines two radicals: a bilinear form radical and a quadratic form radical (cf. section 2 of notes of Bill Casselman, p. 3): $$ \text{rad}_{\beta}=\{m\in M:\forall m'\in M:\beta(m,m')=0\} \ \ \ \text{and} \ \ \ \text{rad}_{q}=\{m\in M:q(m)=0\}. $$ If $\text{rad}_{\beta}=0$, we call $q$ (strictly) non-degenerate, if $\text{rad}_{q}=0$, we call $q$ regular (or weakly non-degenerate). As a fact non-degenerate implies regular and if $2\in R^{\times}$, i.e. $2$ is invertible in $R$, regular implies non-degenerate, in which case $\text{rad}_{\beta}=\text{rad}_{q}$. In general there always exists an orthogonal splitting $M=\text{rad}_{q}\oplus^\bot U$, where $U$ is a quadratic submodule of $M$ that is regular - cf. notes of Bill Casselman, p. 4.
Thus the classification of quadratic modules reduces to classifying regular quadratic modules.
The classification of regular quadratic modules is well known in many cases, e.g. for
local fields (cf. Lam "Introduction to quadratic forms over fields", ch. VI.2 or Corollary 2.8 of notes of Manuel Araújo for an explicit account for $p$-adic numbers)
finite fields (including the characteristic $2$ case: cf. other notes of Bill Casselman, Corollary 5.4, p. 10)
$p$-adic integers (cf. Kitaoka, "Arithmetic of Quadratic Forms", ch. 5.3)
It seems that the study over rings $\mathbb{Z}/n\mathbb{Z}$ can be reduced to rings of type $\mathbb{Z}/p^k\mathbb{Z}$, cf. notes of Manuel Araújo Lemma 3.18 for one direction and Proposition 3.21 for the other direction in case $4\nmid n$.
The rings $\mathbb{Z}/p^k\mathbb{Z}$ are somewhat intermediate between prime finite fields $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ and $p$-adic integers $\mathbb{Z}_p=\lim\limits_{\stackrel{\longleftarrow}{k}}\mathbb{Z}/p^k\mathbb{Z}$, so the classification of regular quadratic forms should be known in this case, too.
Indeed the notes of Manuel Araújo, Theorem 3.5 give a classification for $p$ odd (and in Theorem 3.16 for $p=2$ and $k=1$, which is the $\mathbb{F}_2$-case).
Question: What is the (explicit) classification of regular quadratic modules over the rings $R=\mathbb{Z}/n\mathbb{Z}$? What is the (explicit) classification of regular quadratic modules over the rings $R=\mathbb{Z}/2^k\mathbb{Z}$ and is a statement similar to Lemma 3.18 from the notes of Manuel Araújo valid for general $n$ without the $4$-divisibility condition?
Remarks: The proof of Theorem 3.5 notes of Manuel Araújo for the odd $p$ case makes use of several things, that are different (harder?) for $p=2$:
A. all quadratic forms admit a "diagonalized representation",
B. Squares have a simple uniform group structure $\frac{\left(\mathbb{Z}/p^k\mathbb{Z}\right)^{\times}}{\left(\left(\mathbb{Z}/p^k\mathbb{Z}\right)^{\times}\right)^2}\cong\mathbb{Z}/2\mathbb{Z}$.
C. Hensel lifting applies straightforward to quadratic polynomials.
For $p=2$ we have:
A. "diagonalized representation" do simpliy not exist in general (cf. disscusion on p. 17 of the notes of Manuel Araújo for examples over $\mathbb{F}_2$).
B. $\frac{\left(\mathbb{Z}/2^k\mathbb{Z}\right)^{\times}}{\left(\left(\mathbb{Z}/2^k\mathbb{Z}\right)^{\times}\right)^2}\cong\mathbb{Z}/l_k\mathbb{Z}$, where $l_1=1, l_2=2$ and $l_k=4$ for $k\geq 3$.
C. Hensel lifting is not immediately clear, but the note offers an alternative route that should make the need for Hensel lifting superfluous.
This is not a complete answer to my own question. I did some computer experiments for small $n$ and $d=\text{rk}_RM$, which I think shed some light on the complexity of the problem, that might be interesting enough to share, in particular since my search of the literature did not turn up a lot.
Although not completely congruent with my question, I found some material dealing with generalizations of quadratic forms of the form $Q:\left(\mathbb{Z}/2\mathbb{Z}\right)^d\rightarrow\mathbb{Z}/4\mathbb{Z}$ which seem to have topological applications: a paper by Kai-Uwe Schmidt or this paper by Jay A. Wood might serve as starting points.
Method: encoding and algorithm
The computer search was done using this encoding: any quadratic form can be represented 1-to-1 by an upper triangular matrix. E.g. for $p=2$, $n=q=p^3$ and $R=\mathbb{Z}/2^3\mathbb{Z}$ we have for the rank $d=3$ module $M=R^3=\bigoplus\limits_{i=1}^3Rx_i$ the quadratic form $Q$ and its encoding (by abusing notation also denoted $Q$): $$ Q(x_1,x_2,x_3)=x_1^2+2x_1x_2+4x_1x_3+3x_2^2+5x_2x_3+5x_3^2 \rightsquigarrow Q=\begin{pmatrix}1&2&4\\0&3&5\\0&0&5\end{pmatrix}. $$ A change of coordinates of a free, rank-$d$ $R$-module can be identified with an invertible matrix $M\in\text{GL}_d(\mathbb{Z}/n\mathbb{Z})$, e.g. for our example: $$ [x_1=y_1+3y_2+2y_3,\quad x_2=3y_1+4y_2,\quad x_3=5y_3]\rightsquigarrow M=\begin{pmatrix}1&3&2\\3&4&0\\0&0&5\end{pmatrix}. $$ Applying the transformation in matrix forms means to "transpose-conjugate" $Q$ by $M$, in the example: $$ M^\top QM=\begin{pmatrix}1&3&0\\3&4&0\\2&0&5\end{pmatrix}\begin{pmatrix}1&2&4\\0&3&5\\0&0&5\end{pmatrix}\begin{pmatrix}1&3&2\\3&4&0\\0&0&5\end{pmatrix}=\begin{pmatrix}2&7&1\\1&1&6\\6&6&1\end{pmatrix}=\hat{Q}. $$ However, the naïve transform $\hat{Q}$ is not in upper triangular form. We form a repaired transform $\tilde{Q}$ by setting $\tilde{Q}_{i,j}=\begin{cases} 0,&\text{ if }i>j\\ \hat{Q}_{i,i},&\text{ if }i=j\\ \hat{Q}_{i,j}+\hat{Q}_{j,i},&\text{ if }i<j\end{cases}$. The arithmetic is in $R$. In the example: $$ \tilde{Q}=\begin{pmatrix}2&0&7\\0&1&4\\0&0&1\end{pmatrix} \leftrightsquigarrow \tilde{Q}(y_1,y_2,y_3)=2y_1^2+7y_1y_3+y_2^2+4y_2y_3+y_3^2. $$ This example also illustrates that the discriminant (sometimes called determinant) is not a good invariant in characteristic $p=2$: $\det(Q)=7\in R^{\times}$, while $\det(\tilde{Q})=2\not\in R^{\times}$. Not only are square classes not respected, but even not invertibility or being a zero-divisor. An even more disturbing well-known example over the field $R=\mathbb{Z}/2\mathbb{Z}$ can be found in the notes of Manuel Araújo, Lemma 3.12, p. 17: $$ Q=\begin{pmatrix}1&1&&\\0&1&&\\&&1&1\\&&0&1\end{pmatrix}\sim \begin{pmatrix}0&1&&\\0&0&&\\&&0&1\\&&0&0\end{pmatrix}=\tilde{Q}, $$ where $\det(Q)=1$ is the unit in $R$ and $\det(\tilde{Q})=0$ is the zero in $R$.
In $\text{char}\neq 2$ the same would also occur, if we identified quadratic forms with upper triangular matrices. E.g. over $R=\mathbb{Z}/3\mathbb{Z}$ the form symmetrically represented by $Q=\begin{pmatrix}1&1\\1&0\end{pmatrix}$ has discriminant $\text{disc}(Q)=-1=2\in R^{\times}\setminus\left(R^{\times}\right)^2$, while its triangularized form $\tilde{Q}=\begin{pmatrix}1&2\\0&0\end{pmatrix}$ has $\det(\tilde{Q})=0\not\in R^{\times}$. The discriminant is only a well-defined invariant for quadratic forms, when a symmetrized representation is available. However in the $\text{char}=2$-case there is no good correspondence between quadratic forms and symmetric matrices. Similarly for $2\mid n$.
The computer search for the classification of quadratic forms on a rank$=d$-module $M=R^d$ works in principle like this: enumerate all quadratic forms by the set $\mathcal{Q}$ of upper triangular $d\times d$-matrices over $R$. Enumerate all invertible matrices to encode all allowed coordinate changes for equivalence in a set $\text{GL}_d(R)$. For each $M\in\text{GL}_d(R)$ and $Q\in\mathcal{Q}$ let the computer compute the (upper-triangular-repaired) action by "transpose-conjugation" $Q.M=\tilde{Q}\in\mathcal{Q}$ and identify all equivalence classes. Of course, some tricks are helpful to increase efficiency and reduce computational resource use.
The field case $R=\mathbb{Z}/p\mathbb{Z}$ for $p$ a prime
We test the implementation with this well-known case. We use the variables as introduced above. We give the equivalence classes by a set of representative matrices. Omitted matrix entries are always understood as $0$. We first state some interesting $p$-dependent (in-)congruences (note that the "smallest" quadratic residues in terms of positive integers are $2$ for $p\in\{3,5\}$ and $3$ for $p\in\{7\}$):
$p=3$: $\begin{pmatrix}0&1\\&0\end{pmatrix}\sim\begin{pmatrix}1&0\\&2\end{pmatrix}$, $p=5$: $\begin{pmatrix}0&1\\&0\end{pmatrix}\sim\begin{pmatrix}1&0\\&1\end{pmatrix}$, $p=7$: $\begin{pmatrix}0&1\\&0\end{pmatrix}\sim\begin{pmatrix}1&0\\&3\end{pmatrix}$, $p=2$: $\begin{pmatrix}0&1\\&0\end{pmatrix}\not\sim\begin{pmatrix}1&0\\&1\end{pmatrix}$.
Also note that for $p=3$: $\begin{pmatrix}0&0&1\\&2&0\\&&0\end{pmatrix}\sim\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix}$, $\begin{pmatrix}0&1&&\\&0&&\\&&0&1\\&&&0\end{pmatrix}\sim\begin{pmatrix}1&0&&\\&2&&\\&&1&0\\&&&2\end{pmatrix}\sim\begin{pmatrix}1&0&&\\&1&&\\&&1&0\\&&&1\end{pmatrix}$.
$R=\mathbb{Z}/3\mathbb{Z}$
$d=1$: $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix}$
$d=2$: $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix}$
$d=3$: $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix}$
$d=4$: $\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&0\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&0\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&1&0\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&1&0\\&&&1\end{pmatrix},\begin{pmatrix}1&0&0&0\\&1&0&0\\&&1&0\\&&&1\end{pmatrix},\begin{pmatrix}1&0&0&0\\&1&0&0\\&&1&0\\&&&2\end{pmatrix}$
No surprises here. The representatives are listed with increasing quadratic form rank. For each free module rank $d$ we get exactly two non-degenerate full rank classes of quadratic forms. For each possible rank $r$ of a quadratic form with $0<r<d$ (note the difference between quadratic form rank and free module rank) we unsurprisingly also get two classes of quadratic forms that can be represented by an orthogonal direct sum of a $0$-form and a non-degenerate form in lower dimensions. For $r=0$ we get exactly one form: the $0$-form on $M$.
Fun fact: for $d=2$ the two full rank classes contain a different number of distinct forms: $\lvert\left[\begin{pmatrix}0&1\\&0\end{pmatrix}\right]\rvert = 12>6= \lvert\left[\begin{pmatrix}1&0\\&2\end{pmatrix}\right]\rvert$.
We give a list of class sizes in the order the representatives are listed above:
For $d=1$: 1,1,1
For $d=2$: 1,4,4,12,6
For $d=3$: 1,13,13,156,78,234,234
For $d=4$: 1,40,40,1560,780,9360,9360,21060,16848
$R=\mathbb{Z}/5\mathbb{Z}$
$d=1$: $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix}$
$d=2$: $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix}$
$d=3$: $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix}$
This follows the same pattern as for $p=3$ in accordance with established results. Class sizes:
For $d=1$: 1,2,2
For $d=2$: 1,12,12,60,40
For $d=3$: 1,62,62,1860,1240,6200,6200
$R=\mathbb{Z}/7\mathbb{Z}$
$d=1$: $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}3\end{pmatrix}$
$d=2$: $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix}$
$d=3$: $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix}$
This follows the same pattern as for $p=3$ in accordance with established results. Class sizes:
For $d=1$: 1,3,3
For $d=2$: 1,24,24,168,126
For $d=3$: 1,171,171,9576,7182,50274,50274
$R=\mathbb{Z}/2\mathbb{Z}$
$d=1$: $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix}$
$d=2$: $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&1\\&0\end{pmatrix},\begin{pmatrix}1&1\\&1\end{pmatrix}$
$d=3$: $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&0&1\\&&0\end{pmatrix}$
$d=4$: $\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&1\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&1&0&0\\&0&0&0\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&1&0&0\\&0&0&0\\&&1&1\\&&&1\end{pmatrix}$
$d=5$: $\begin{pmatrix}0&&&&\\&0&&&\\&&0&&\\&&&0&\\&&&&0\end{pmatrix},\begin{pmatrix}0&&&&\\&0&&&\\&&0&&\\&&&0&\\&&&&1\end{pmatrix},\begin{pmatrix}0&&&&\\&0&&&\\&&0&&\\&&&0&1\\&&&&0\end{pmatrix},\begin{pmatrix}0&&&&\\&0&&&\\&&0&&\\&&&1&1\\&&&&1\end{pmatrix},\begin{pmatrix}0&&&&\\&0&&&\\&&1&0&0\\&&&0&1\\&&&&0\end{pmatrix},\begin{pmatrix}0&&&&\\&0&1&0&0\\&&0&0&0\\&&&0&1\\&&&&0\end{pmatrix},\begin{pmatrix}0&&&&\\&0&1&0&0\\&&0&0&0\\&&&1&1\\&&&&1\end{pmatrix},\begin{pmatrix}1&0&0&0&0\\&0&1&0&0\\&&0&0&0\\&&&0&1\\&&&&0\end{pmatrix}$
For each module rank we have the expected number of full rank regular classes: one if $2\nmid d$ and two if $2\mid d$. Again the computer did not pick representatives in standard form, but they are easily matched; the full rank representatives can always be found at the end of the list. As before for $p\neq 2$ all remaining classes are "lifts" from lower module ranks by forming direct sums with suitable $0$-forms.
Class sizes:
For $d=1$: 1,1
For $d=2$: 1,3,3,1
For $d=3$: 1,7,21,7,28
For $d=4$: 1,15,105,35,420,280,168
For $d=5$: 1,31,465,155,4340,8680,5208,13888
The prime power case $R=\mathbb{Z}/q\mathbb{Z}$ for $q=p^2$ a prime square
In contrast to the notes of Manuel Araújo we include degenerate quadratic forms. We expect the same classification independent of $p$ if $p\neq 2$. For $p\neq 2$ the data supports that all quadratic forms are diagonalizable (even degenerate ones) and that the conjectural formula for the number of classes is: $1+4\cdot\frac{d(d+1)}{2}$. We list representatives of the forms ascending to rank, within rank by number of orthogonal factors starting with most factors and sorting by number of degenerate factors starting with least degeneracy. Again we give class sizes.
$R=\mathbb{Z}/3^2\mathbb{Z}$
$d=1$ ($5$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}6\end{pmatrix}$
$d=2$ ($13$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}0&\\&6\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&6\end{pmatrix},\begin{pmatrix}2&0\\&3\end{pmatrix},\begin{pmatrix}2&0\\&6\end{pmatrix},\begin{pmatrix}3&0\\&3\end{pmatrix},\begin{pmatrix}3&0\\&6\end{pmatrix}$
$d=3$ ($25$ classes): $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&2&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&2&0\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&2&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&2&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&6\end{pmatrix},\begin{pmatrix}2&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}2&0&0\\&3&0\\&&6\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&6\end{pmatrix}$
Class sizes:
For $d=1$: 1,3,3,1,1
For $d=2$: 1,36,36,4,4,162,324,36,36,36,36,6,12
For $d=3$: 1,351,351,13,13,18954,37908,1404,1404,1404,1404,78,156,170586,170586,18954,18954,37908,37908,2106,4212,2106,4212,234,234
$R=\mathbb{Z}/5^2\mathbb{Z}$
$d=1$ ($5$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix},\begin{pmatrix}5\end{pmatrix},\begin{pmatrix}10\end{pmatrix}$
$d=2$ ($13$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}0&\\&5\end{pmatrix},\begin{pmatrix}0&\\&10\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&5\end{pmatrix},\begin{pmatrix}1&0\\&10\end{pmatrix},\begin{pmatrix}2&0\\&5\end{pmatrix},\begin{pmatrix}2&0\\&10\end{pmatrix},\begin{pmatrix}5&0\\&5\end{pmatrix},\begin{pmatrix}5&0\\&10\end{pmatrix}$
Class sizes:
For $d=1$: 1,10,10,2,2
For $d=2$: 1,300,300,12,12,7500,5000,600,600,600,600,60,40
$R=\mathbb{Z}/7^2\mathbb{Z}$
$d=1$ ($5$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}7\end{pmatrix},\begin{pmatrix}21\end{pmatrix}$
$d=2$ ($13$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}0&\\&7\end{pmatrix},\begin{pmatrix}0&\\&21\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&7\end{pmatrix},\begin{pmatrix}1&0\\&21\end{pmatrix},\begin{pmatrix}3&0\\&7\end{pmatrix},\begin{pmatrix}3&0\\&21\end{pmatrix},\begin{pmatrix}7&0\\&7\end{pmatrix},\begin{pmatrix}7&0\\&21\end{pmatrix}$
Class sizes:
For $d=1$: 1,21,21,3,3
For $d=2$: 1,1176,1176,24,24,43218,57624,3528,3528,3528,3528,126,168
$R=\mathbb{Z}/2^2\mathbb{Z}$
For $d=4$ and forms rank $4$ we did not attempt to find diagonalizations and representatives with many orthogonal factors. These representatives are given in the order and form determined by the enumeration of the algorithm used.
$d=1$ ($4$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}2\end{pmatrix}$
$d=2$ ($12$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}3&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&1\\&1\end{pmatrix},\begin{pmatrix}0&1\\&0\end{pmatrix},\begin{pmatrix}2&2\\&2\end{pmatrix},\begin{pmatrix}0&2\\&0\end{pmatrix}$
$d=3$ ($22$ classes): $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&2&2\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&0&2\\&&0\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}3&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}2&0&0\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}2&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}2&0&0\\&0&2\\&&0\end{pmatrix}$
$d=4$ ($39$ classes): $\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&\\&&&2\end{pmatrix}, \begin{pmatrix}0&&&\\&0&&\\&&1&0\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&0\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&3&0\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&0\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&1&1\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&2&2\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&0&&\\&&0&2\\&&&0\end{pmatrix}, \begin{pmatrix}0&&&\\&1&0&0\\&&1&0\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&1&0\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&3&0\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&3&0&0\\&&3&0\\&&&3\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&1&0\\&&&2\end{pmatrix},\begin{pmatrix}0&&&\\&1&0&0\\&&1&1\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&3&0&0\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&2&0&0\\&&1&1\\&&&1\end{pmatrix},\begin{pmatrix}0&&&\\&2&0&0\\&&0&1\\&&&0\end{pmatrix},\begin{pmatrix}0&&&\\&2&0&0\\&&0&2\\&&&0\end{pmatrix}, \begin{pmatrix}0&0&0&1\\&0&1&0\\&&0&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&0&1&0\\&&0&0\\&&&0\end{pmatrix},\begin{pmatrix}1&0&0&2\\&0&1&0\\&&0&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&0&2&0\\&&0&0\\&&&0\end{pmatrix},\begin{pmatrix}1&0&0&2\\&0&2&0\\&&0&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&1\\&1&0&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&1&0&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&1\\&2&0&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&2&0&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&1\\&1&1&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&1&1&0\\&&1&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&1\\&2&2&0\\&&2&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&2&2&0\\&&2&0\\&&&0\end{pmatrix},\begin{pmatrix}1&0&0&2\\&2&2&0\\&&2&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&1\\&3&2&0\\&&2&0\\&&&0\end{pmatrix},\begin{pmatrix}0&0&0&2\\&3&2&0\\&&2&0\\&&&0\end{pmatrix},\begin{pmatrix}1&1&1&1\\&1&0&0\\&&1&0\\&&&1\end{pmatrix}$
Class sizes:
For $d=1$: 1,1,1,1
For $d=2$: 1,3,3,3,3,6,3,6,8,24,1,3
For $d=3$: 1,7,7,7,21,42,21,42,224,672,7,21,28,84,84,28,84,896,896,224,672,28
For $d=4$: 1,15,15,15,105,210,105,210,4480,13440,35,105, 420,1260,1260,420,1260,53760,53760,13440,40320,420, 286720,40320,107520,280,2520,53760,1680,107520,3360,172032,13440,13440,168,840,53760,1680,4480
We note some interesting congruences (mostly surprising diagonalizations).
For $d=2$: $\begin{pmatrix}1&2\\&0\end{pmatrix}\sim\begin{pmatrix}1&0\\&3\end{pmatrix}$,$\begin{pmatrix}3&2\\&2\end{pmatrix}\sim\begin{pmatrix}3&0\\&3\end{pmatrix}$
For $d=3$: $\begin{pmatrix}1&0&0\\&0&2\\&&0\end{pmatrix}\sim\begin{pmatrix}1&0&0\\&1&0\\&&3\end{pmatrix}$,$\begin{pmatrix}3&0&0\\&0&2\\&&0\end{pmatrix}\sim\begin{pmatrix}1&0&0\\&3&0\\&&3\end{pmatrix}$,$\begin{pmatrix}1&0&0\\&2&2\\&&2\end{pmatrix}\sim\begin{pmatrix}3&0&0\\&3&0\\&&3\end{pmatrix}$,$\begin{pmatrix}1&0&0\\&0&1\\&&0\end{pmatrix}\sim\begin{pmatrix}1&0&0\\&1&1\\&&1\end{pmatrix}$,$\begin{pmatrix}1&0&0\\&3&0\\&&2\end{pmatrix}\sim\begin{pmatrix}1&2&0\\&0&0\\&&2\end{pmatrix}\sim\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix}$,$\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix}\sim\begin{pmatrix}0&&\\&2&0\\&&2\end{pmatrix}\sim\begin{pmatrix}2&0&0\\&2&0\\&&2\end{pmatrix}$
The prime power case $R=\mathbb{Z}/q\mathbb{Z}$ for $q=p^3$ a prime cube
Again degenerate quadratic forms are included. From the non-degenerate case we know to expect essentially the same classification independent of $p$ if $p\neq 2$. The data supports this and also that all quadratic forms are diagonalizable (even degenerate ones) and that the conjectural formula for the number of classes is: $1+6\cdot\frac{d(d+1)(d+2)}{6}$. For $p\neq 2$ and $q=p^n$ for arbitrary $n$ we conjecture the number of classes in (module) rank $d$ to be: $1+2n\begin{pmatrix}d+n\\n\end{pmatrix}$. We follow the same conventions as before.
$R=\mathbb{Z}/3^3\mathbb{Z}$
$d=1$ ($7$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}6\end{pmatrix},\begin{pmatrix}9\end{pmatrix},\begin{pmatrix}18\end{pmatrix}$
$d=2$ ($25$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}0&\\&6\end{pmatrix},\begin{pmatrix}0&\\&9\end{pmatrix},\begin{pmatrix}0&\\&18\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&6\end{pmatrix},\begin{pmatrix}1&0\\&9\end{pmatrix},\begin{pmatrix}1&0\\&18\end{pmatrix},\begin{pmatrix}2&0\\&3\end{pmatrix},\begin{pmatrix}2&0\\&6\end{pmatrix},\begin{pmatrix}2&0\\&9\end{pmatrix},\begin{pmatrix}2&0\\&18\end{pmatrix},\begin{pmatrix}3&0\\&3\end{pmatrix},\begin{pmatrix}3&0\\&6\end{pmatrix},\begin{pmatrix}3&0\\&9\end{pmatrix},\begin{pmatrix}3&0\\&18\end{pmatrix},\begin{pmatrix}6&0\\&9\end{pmatrix},\begin{pmatrix}6&0\\&18\end{pmatrix},\begin{pmatrix}9&0\\&9\end{pmatrix},\begin{pmatrix}9&0\\&18\end{pmatrix}$
Class sizes:
For $d=1$: 1,9,9,3,3,1,1
For $d=2$: 1,324,324,36,36,4,4,4374,8748,972,972,324,324,972,972,324,324,162,324,36,36,36,36,6,12
$R=\mathbb{Z}/5^3\mathbb{Z}$
$d=1$ ($7$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix},\begin{pmatrix}5\end{pmatrix},\begin{pmatrix}10\end{pmatrix},\begin{pmatrix}25\end{pmatrix},\begin{pmatrix}50\end{pmatrix}$
$d=2$ ($25$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}0&\\&5\end{pmatrix},\begin{pmatrix}0&\\&10\end{pmatrix},\begin{pmatrix}0&\\&25\end{pmatrix},\begin{pmatrix}0&\\&50\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&5\end{pmatrix},\begin{pmatrix}1&0\\&10\end{pmatrix},\begin{pmatrix}1&0\\&25\end{pmatrix},\begin{pmatrix}1&0\\&50\end{pmatrix},\begin{pmatrix}2&0\\&5\end{pmatrix},\begin{pmatrix}2&0\\&10\end{pmatrix},\begin{pmatrix}2&0\\&25\end{pmatrix},\begin{pmatrix}2&0\\&50\end{pmatrix},\begin{pmatrix}5&0\\&5\end{pmatrix},\begin{pmatrix}5&0\\&10\end{pmatrix},\begin{pmatrix}5&0\\&25\end{pmatrix},\begin{pmatrix}5&0\\&50\end{pmatrix},\begin{pmatrix}10&0\\&25\end{pmatrix},\begin{pmatrix}10&0\\&50\end{pmatrix},\begin{pmatrix}25&0\\&25\end{pmatrix},\begin{pmatrix}25&0\\&50\end{pmatrix}$
Class sizes:
For $d=1$: 1,50,50,10,10,2,2
For $d=2$: 1,7500,7500,300,300,12,12,937500, 625000,75000,75000,15000,15000,75000,75000,15000,15000,7500,5000,600,600,600,600,60,40
$R=\mathbb{Z}/7^3\mathbb{Z}$
$d=1$ ($7$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}7\end{pmatrix},\begin{pmatrix}21\end{pmatrix},\begin{pmatrix}49\end{pmatrix},\begin{pmatrix}147\end{pmatrix}$
Class sizes:
For $d=1$: 1,147,147,21,21,3,3
$R=\mathbb{Z}/2^3\mathbb{Z}$
$d=1$ ($8$ classes): $\begin{pmatrix}0\end{pmatrix},\begin{pmatrix}1\end{pmatrix},\begin{pmatrix}2\end{pmatrix},\begin{pmatrix}3\end{pmatrix},\begin{pmatrix}4\end{pmatrix},\begin{pmatrix}5\end{pmatrix},\begin{pmatrix}6\end{pmatrix},\begin{pmatrix}7\end{pmatrix}$
$d=2$ ($30$ classes): $\begin{pmatrix}0&\\&0\end{pmatrix},\begin{pmatrix}0&\\&1\end{pmatrix},\begin{pmatrix}0&\\&2\end{pmatrix},\begin{pmatrix}0&\\&3\end{pmatrix},\begin{pmatrix}0&\\&4\end{pmatrix},\begin{pmatrix}0&\\&5\end{pmatrix},\begin{pmatrix}0&\\&6\end{pmatrix},\begin{pmatrix}0&\\&7\end{pmatrix}, \begin{pmatrix}0&1\\&0\end{pmatrix},\begin{pmatrix}0&2\\&0\end{pmatrix},\begin{pmatrix}1&0\\&7\end{pmatrix},\begin{pmatrix}0&4\\&0\end{pmatrix},\begin{pmatrix}1&0\\&4\end{pmatrix},\begin{pmatrix}2&0\\&6\end{pmatrix},\begin{pmatrix}3&0\\&4\end{pmatrix},\begin{pmatrix}1&0\\&1\end{pmatrix},\begin{pmatrix}1&0\\&2\end{pmatrix},\begin{pmatrix}1&0\\&3\end{pmatrix},\begin{pmatrix}1&0\\&5\end{pmatrix},\begin{pmatrix}1&0\\&6\end{pmatrix},\begin{pmatrix}1&1\\&1\end{pmatrix},\begin{pmatrix}2&0\\&2\end{pmatrix},\begin{pmatrix}3&0\\&2\end{pmatrix},\begin{pmatrix}2&0\\&4\end{pmatrix},\begin{pmatrix}5&0\\&2\end{pmatrix},\begin{pmatrix}2&2\\&2\end{pmatrix},\begin{pmatrix}3&0\\&7\end{pmatrix},\begin{pmatrix}3&0\\&3\end{pmatrix},\begin{pmatrix}4&4\\&4\end{pmatrix},\begin{pmatrix}6&0\\&6\end{pmatrix}$
$d=3$ ($74$ classes): $\begin{pmatrix}0&&\\&0&\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&4\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&5\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&0&\\&&7\end{pmatrix}, \begin{pmatrix}0&&\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&5\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&7\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&1&0\\&&4\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&7\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&5&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&3&0\\&&4\end{pmatrix},\begin{pmatrix}0&&\\&2&0\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&2&0\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&6&0\\&&6\end{pmatrix},\begin{pmatrix}0&&\\&2&0\\&&4\end{pmatrix},\begin{pmatrix}0&&\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}0&&\\&0&2\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&2&2\\&&2\end{pmatrix},\begin{pmatrix}0&&\\&0&4\\&&0\end{pmatrix},\begin{pmatrix}0&&\\&4&4\\&&4\end{pmatrix}, \begin{pmatrix}1&0&0\\&1&0\\&&1\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&5\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&7\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&1&0\\&&4\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&7\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&3&0\\&&4\end{pmatrix},\begin{pmatrix}1&0&0\\&5&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&2&0\\&&2\end{pmatrix},\begin{pmatrix}1&0&0\\&2&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&6&0\\&&6\end{pmatrix},\begin{pmatrix}1&0&0\\&2&0\\&&4\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&3\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&7\end{pmatrix},\begin{pmatrix}3&0&0\\&3&0\\&&4\end{pmatrix},\begin{pmatrix}3&0&0\\&2&0\\&&2\end{pmatrix},\begin{pmatrix}2&0&0\\&2&0\\&&2\end{pmatrix},\begin{pmatrix}2&0&0\\&2&0\\&&6\end{pmatrix},\begin{pmatrix}2&0&0\\&6&0\\&&6\end{pmatrix},\begin{pmatrix}6&0&0\\&6&0\\&&6\end{pmatrix},\begin{pmatrix}2&0&0\\&2&0\\&&4\end{pmatrix},\begin{pmatrix}1&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}3&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}5&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}7&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}1&0&0\\&0&4\\&&0\end{pmatrix},\begin{pmatrix}3&0&0\\&0&4\\&&0\end{pmatrix},\begin{pmatrix}5&0&0\\&0&4\\&&0\end{pmatrix},\begin{pmatrix}7&0&0\\&0&4\\&&0\end{pmatrix},\begin{pmatrix}2&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}6&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}2&0&0\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}6&0&0\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}2&0&0\\&0&2\\&&0\end{pmatrix},\begin{pmatrix}6&0&0\\&0&2\\&&0\end{pmatrix},\begin{pmatrix}4&0&0\\&0&1\\&&0\end{pmatrix},\begin{pmatrix}4&0&0\\&1&1\\&&1\end{pmatrix},\begin{pmatrix}4&0&0\\&0&2\\&&0\end{pmatrix},\begin{pmatrix}4&0&0\\&2&2\\&&2\end{pmatrix},\begin{pmatrix}4&0&0\\&0&4\\&&0\end{pmatrix}$
Class sizes:
For $d=1$: 1,1,1,1,1,1,1,1
For $d=2$: 1,6,3,6,3,6,3,6, 192, 24, 24, 3, 12, 6, 12, 12, 12, 24, 12, 12, 64, 3, 12, 6, 12, 8, 12, 12, 1, 3
For $d=3$: 1,28,7,28,7,28,7,28, 336,672,336,672,168,168,168,336,336,168,168,168,21,42,21,42,21504,7168,672,224,21,7, 896,2688,896,2688,1344,1344,672,2688,2688,1344,1344,1344,336,336,336,672,896,896,672,336,28,84,84,28,84,28672,28672,28672,28672,112,112,112,112,21504,21504,7168,7168,896,896,21504,7168,672,224,28
To be continued