I have 2 groups, generated by 4x4 matrices such as X (x+1,y,z), Y (x,y+1,z), Z(x,y,z+1), R (-x,-y,z), F(-y,x,z). R - 2-fold rotation matrix and F+ - 4-fold rotation matrix (counterclockwise). Each 4x4 matrix consists of rotating matrix and translation vector in 3D-space, written in one matrix. For example: $$ F=\begin{pmatrix}0&-1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$$ and $$ X=\begin{pmatrix}1&0&0&1\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}$$ First group is generated from all 5 matrices (it's Fedorov group #75) Second is generated from only X, Z and F. Because $Y=F^{-1} * X * F$ and $R=F^{2}$ we can say that they are equal. How can i automate process of evaluating any matrix from set of others (if it's possible), for example: try to solve Y as equation of X, Z, F. OR How can i compare 2 groups, generated with different generating set, is it possible to do in any linear algebra systems, such as GAP?
2026-03-28 12:38:29.1774701509
question about space groups
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Comparing two infinite groups is a very hard problem. If you know that one of them is crystallographic, then GAP has packages Cryst - Computing with crystallographic groups and CrystCat - The crystallographic groups catalog. It might happen that these may help here, but I suggest to ask this in the GAP Forum or GAP Support to reach package authors
(Just copied here my earlier comment to remove this question from the unanswered queue).