Beyond the syllabusSymmetry lab
SYMMETRY EXTENSION / INTERACTIVE EXPERIENCE

One cube, four actions

Explore the action3D · EXACT FINITE ACTION
xyz2134 Drag to orbit · select a label
Apply generator
Q=I=(100010001)Q=I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
πX(Q)=()\pi_X(Q)=()det Q = 1 · 4 cycles
48|G|
4|Orb(1)|
12|Stab(1)|
2|ker π|

Every unoriented diagonal is reachable. Each destination in this orbit is reached by exactly 12 group elements: 48=4⋅1248=4\cdot 12. Inversion fixes all four diagonal lines, so the kernel is {I, −I}.

Explore the elements
Count colorings with Burnside’s lemma#({1,…,k}X/G)=148∑g∈Gkc(g)\#(\{1,\ldots,k\}^X/G)=\frac{1}{48}\sum_{g\in G}k^{c(g)}

For 3 colors on diagonals, there are 15 inequivalent colorings. Repeated colors are allowed. A coloring fixed by g must be constant on every cycle.

2 elements × 4 cycles12 elements × 3 cycles22 elements × 2 cycles12 elements × 1 cycles
From this action to a permutation representation

Attach one basis vector eᵢ to each object. The action permutes coefficients by PQei=eπ(Q)(i)P_Qe_i=e_{\pi(Q)(i)}. This representation has dimension 4; the geometric matrix Q has dimension 3.

PQ=(1000010000100001)P_Q=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}χperm(Q)=#Fix⁡X(Q)=4,tr⁡Q=3\chi_{\rm perm}(Q)=\#\operatorname{Fix}_X(Q)=4,\quad\operatorname{tr}Q=3

These are different representations of the same group. A permutation character counts fixed basis labels; the geometric trace measures the action on ambient coordinates.

Generators and composition convention

Column vectors; AB applies B first. Every button left-multiplies Q. The matrix product is exact; camera movement is independent of the group action.

A=(10000−1010),B=(0−10100001)A=\begin{pmatrix}1&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad B=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix}J=−I3,A4=B4=J2=I,JQ=QJJ=-I_3,\qquad A^4=B^4=J^2=I,\quad JQ=QJ

A and B generate all 24 rotations. Adjoining J yields 48 symmetries. Determinant −1 includes inversion and rotoreflections, not only plane reflections.