If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $\tau$, and finally join corresponding points of the two copies, then we obtain a framework with `extrusion' symmetry in the direction of $\tau$. This process may be repeated $t$ times to obtain a framework whose underlying graph has $\mathbb{Z}_2^t$ as a subgroup of its automorphism group and which has `$t$-fold extrusion' symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extruded frameworks.

In this talk we show that while $t$-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with $t$-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. In particular,
this allows us to use Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to
the more general point-hyperplane frameworks with $t$-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the
infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions.

(This is joint work with John Owen.)