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The 15 puzzle in 2d is associated with the Alternating group A15. https://en.wikipedia.org/wiki/15_puzzle

Rubik's has produced the Rubik's slide: a 3d slide puzzle. https://www.rubiks.com/en-uk/rubiks-slide.html

And here's a Youtube video of the Rubik's slide being solved: https://www.youtube.com/watch?v=o4JupVA5zfI

Does it similarly have an associated group? What about an n x n x n-1 3d puzzle -what group would it have?

Not to be confused with an earlier electronic puzzle also called Rubik's slide.

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    $\begingroup$ The gap starts in a corner of the cube, and effectively all slides move it to another corner of the cube. Then there are three legal slides from any position. For each, determine a permutation of the positions of the 26 external cubies generated by making the slide and then rotating the cube to put the corner back where it started. Then the group is the group generated by these three permutations (each chosen arbitrarily from three options) and the permutation corresponding to a rotation by 1/3 around the axis through the gap and the centre of the cube, quotiented by the latter permutation. $\endgroup$
    – Peter Taylor
    Commented Jan 17, 2022 at 12:11
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    $\begingroup$ Obvious optimisation for manual calculation: it only takes one (slide plus rotate cube to put gap back in position) and the rotation by 1/3 to generate the group. $\endgroup$
    – Peter Taylor
    Commented Jan 17, 2022 at 14:08
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    $\begingroup$ If I'm correctly interpreting the results of my calculations with GAP then the group is $(C_2 \times C_2) \rtimes S_{19}$, but take that with some suspicion. $\endgroup$
    – Peter Taylor
    Commented Jan 17, 2022 at 14:31
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    $\begingroup$ I guess it may be even simpler to not allow rotations of the whole cube and just consider the group of "slides" bringing the gap back to its original position (even though depending on the exact rules, this may slightly reduce the positions considered "solvable"). Then there are 19 pieces which move in total (one corner as well as the centers of each face are fixed), and every move is a composition of 3-cycles, so the group is contained in $A_{19}$. Now looking at a few selected moves will give that it's actually $2$-transitive and contains some short cycle, forcing it to be $A_{19}$ exactly. $\endgroup$
    – Joachim König
    Commented Jan 27, 2022 at 16:40

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