Chirality of the hexagonal enantiomorphic space-group pairs

Throughout, $E=\mathbb R^3$, $G_1,\dots,G_{230}$ are the groups of the explicit catalogue spaceGroupCatalog (the group $G_n$ realises the space-group type with ITA number $n$, and the second member of each enantiomorphic pair is the image of the first under the point inversion $x\mapsto -x$; see the definition SpaceGroupCatalog), and $G\sim_+G'$ means that some invertible affine map $\varphi$ of $E$ with $\det\varphi_{\mathrm{lin}}>0$ satisfies $\varphi G\varphi^{-1}=G'$ (AffOPEquivalent).

For each of the four enantiomorphic pairs $(n,n')$ among $(169,170)$ ($P6_{1}$ / $P6_{5}$), $(171,172)$ ($P6_{2}$ / $P6_{4}$), $(178,179)$ ($P6_{1}22$ / $P6_{5}22$), $(180,181)$ ($P6_{2}22$ / $P6_{4}22$),

(In Lean the catalogue is indexed from $0$, so $G_n$ is spaceGroupCatalog (n-1).) These are the hexagonal cases of the statement that the eleven enantiomorphic pairs of space-group types are chiral, i.e. that the members of a pair are mirror images of each other but are not related by any orientation-preserving affine map.