A complete irredundant list of the $219$ affine classes of space groups

Work in $E=\mathbb R^3$ with its Euclidean metric, and let $\mathrm{Isom}(E)$ be the group of bijective affine isometries of $E$. Call a subgroup $G\le\mathrm{Isom}(E)$ 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) there are three linearly independent vectors $v_1,v_2,v_3$ such that translation by each $v_i$ belongs to $G$.

For subgroups $G_1,G_2\le\mathrm{Isom}(E)$ write $G_1\sim G_2$ (affinely equivalent) if some invertible affine map $\varphi$ of $E$ satisfies $\varphi G_1\varphi^{-1}=G_2$ as sets of affine maps, and $G_1\sim_{+}G_2$ (orientation-preserving affinely equivalent) if such a $\varphi$ can be chosen with $\det(\varphi_{\mathrm{lin}})>0$.

This statement is the enumeration form of the count of $219$ affine classes: there is a list

of crystallographic groups in $\mathbb R^3$ that is

1. complete: every crystallographic group $G$ satisfies $G_i\sim G$ for some index $i$; and
2. irredundant: $G_i\sim G_j$ forces $i=j$.

Here the comparison map $\varphi$ is an arbitrary invertible affine map, with no constraint on the sign of its determinant, so an enantiomorphic pair of space groups becomes a single class. This is the convention under which the classical count drops from $230$ to $219$.

Combined with the fact that $\sim$ is an equivalence relation, the two conditions give the cardinality of the set of $\sim$-classes, and hence the count $219$.

Formalization Note. The list is given as a function $f$ from Fin 219; CrystallographicGroup 3 and AffinelyEquivalent are the crystallographic-group subtype and the unrestricted affine conjugacy relation of the LeanEval definition bundle, used verbatim.