Master enumeration of the space groups of $\mathbb R^3$: the list of $230$ with its affine and Sohncke bookkeeping

Work in $E=\mathbb R^3$ with its Euclidean metric and let $\mathrm{Isom}(E)$ be the group of bijective affine isometries. A subgroup $G\le\mathrm{Isom}(E)$ is crystallographic if (i) for every $x\in E$ and every $\varepsilon>0$ the set $\{g\in G:\operatorname{dist}(g(x),x)\le\varepsilon\}$ is finite, and (ii) $G$ contains translations by three linearly independent vectors. Write $G_1\sim G_2$ if $\varphi G_1\varphi^{-1}=G_2$ for some invertible affine map $\varphi$, and $G_1\sim_{+}G_2$ if such a $\varphi$ can be chosen with $\det(\varphi_{\mathrm{lin}})>0$. Call $G$ a Sohncke group if every $g\in G$ has $\det(g_{\mathrm{lin}})>0$.

The three classical counts — $230$ space-group types, $219$ affine classes, $65$ Sohncke types — are three readings of one and the same classification, and this statement packages them into a single assertion. It asks for a list

of crystallographic groups of $\mathbb R^3$ together with two pieces of bookkeeping data about that list:

1. The list is a complete irredundant set of representatives for $\sim_{+}$. Every crystallographic group $G$ satisfies $G_i\sim_{+}G$ for some $i$, and $G_i\sim_{+}G_j$ forces $i=j$. This is the Fedorov–Schoenflies count of $230$.

2. A selection $s:\{1,\dots,219\}\to\{1,\dots,230\}$ of representatives for $\sim$. Every index $i$ satisfies $G_{s(k)}\sim G_i$ for some $k$, and $G_{s(k)}\sim G_{s(l)}$ forces $k=l$. Since $\sim$ is coarser than $\sim_{+}$, and every crystallographic group is $\sim_{+}$-equivalent (hence $\sim$-equivalent) to some member of the list, the $219$ selected members form a complete irredundant list for $\sim$: the $11$ enantiomorphic pairs among the $230$ have been merged.

3. An injection $t:\{1,\dots,65\}\to\{1,\dots,230\}$ whose image is exactly the set of Sohncke members of the list, i.e. $G_i$ is a Sohncke group if and only if $i=t(k)$ for some $k$. Because the Sohncke condition is invariant under affine conjugacy, the Sohncke groups form a union of $\sim_{+}$-classes, so these $65$ members form a complete irredundant list of the Sohncke groups up to $\sim_{+}$.

Consequently this single statement implies each of space_groups_OP_classification ($230$), space_groups_affine_classification ($219$) and space_groups_sohncke_classification ($65$); those three deductions are supplied as proof-sketch reductions to this theorem, so the mission's frontier is the present statement alone.

Formalization note. CrystallographicGroup 3 is the subtype of crystallographic subgroups, AffOPEquivalent and AffinelyEquivalent are $\sim_{+}$ and $\sim$, and IsOrientationPreservingIsom g says $\det(g_{\mathrm{lin}})>0$; all are the mission's own definitions.