# The Group of rotations of a cube in space.

Am currently working on the Group of rotations of a cube in space. i have identified the 24 rotational symmetries of a cube that forms a group. And i am kind off trying to show that these symmetries forms a group by computing or creating a composition table of all the 24 elements permuting themselves. These is a tedious job to compute manually or physically. so my question or challenge is if there is any existing program or application or any other method to clearly show that the 24 symmetries forms a Group.

http://www.euclideanspace.com/maths/discrete/groups/categorise/finite/cube/

i came across these site but can't find the program or relate his table to what i have in mind because my symmetries are labelled C1-C24.

• What do you want exactly? If your goal is just to prove that the symmetries form a group, this is fairly easy to do using the definition of a symmetry group, without even needing to know what the 24 elements are. If what you want is a better way of representing the symmetries of a cube, you could use matrices, permutations, or quaternions. – Jim Belk Jun 20 '15 at 23:33
• Indeed, as @JimBelk comments, it is "easy" (from some viewpoint) to argue that the collection of set-maps from any given set to itself, possibly stabilizing various subset, preserving various properties, ... is a group. I'd speculate that your issue is that you're feeling over-obligated to make specific reference to details that are inessential for proving the "group property"... Conceivably this conceptual issue is exactly the issue that you or someone else has (perhaps accidentally) "tasked you with"... :) Clarify? – paul garrett Jun 20 '15 at 23:47
• The cube has 4 diagonals (each joining a pair of opposite vertices, through the center of the cube). Every rotation of the cube acts as a permutation of these 4 diagonals. Do any two distinct rotations give you the same permutation of the diagonals? – Gerry Myerson Jun 21 '15 at 0:15
• comments. proving by the approach of definition is easy@ jim belk. @paul garret i am tasked to clearly show all permutations of all the 24 elements on themselves by using a composition table.(either by a program or ortherwise .) @ Gerry Myerson.Any two distinct permutation of a 2 symmtries results to another symmmetry in the group. thanks y`all . Any reference to materials or sites will be of tremendous help,then maybe i can understand better and come back to answer my question. – GT. Jun 21 '15 at 8:02
• As 3x3 matrices the said rotations have the form: 1) all rows and all columns contain a single non-zero entry. 2) that non-zero entry is $\pm1$. 3) the determinant of the matrix is $+1$ (rotation -> orientation-preserving). It is easy to show that the collection of such matrices forms a group. – Jyrki Lahtonen Jun 21 '15 at 21:35