Stunning Cayley tables from Q8, all order-8 groups, and larger groups like A4 and D5
Some groups have strikingly beautiful multiplication tables. Each color pattern is a fingerprint of its group.
Switch between the highlight modes. In "ij = k rules" mode, the identities ij = k, jk = i and ki = j light up in purple.
| * | 1 | i | j | -1 | k | -k | -i | -j |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | i | j | -1 | k | -k | -i | -j |
| i | i | -1 | k | -i | -j | j | 1 | -k |
| j | j | -k | -1 | -j | i | -i | k | 1 |
| -1 | -1 | -i | -j | 1 | -k | k | i | j |
| k | k | j | -i | -k | -1 | 1 | -j | i |
| -k | -k | -j | i | k | 1 | -1 | j | -i |
| -i | -i | 1 | -k | i | j | -j | -1 | k |
| -j | -j | k | 1 | j | -i | i | -k | -1 |
Some groups produce multiplication tables of extraordinary beauty. The color patterns in a Cayley table are a fingerprint -- no two non-isomorphic groups produce the same pattern. Here we showcase some of the most visually striking and mathematically fascinating examples.
From the elegant quaternion algebra to the surprising variety among order-8 groups, these tables reveal structure that formulas alone cannot convey.
Hamilton's quaternions {±1, ±i, ±j, ±k} form a group of order 8 with the famous rules ij = k, jk = i, ki = j. Switch to "ij = k rules" mode to see these identities highlighted in purple.
Key insight: Q8 is non-abelian (ij ≠ ji) yet every subgroup is normal -- making it a rare "Hamiltonian" group. Its multiplication table has a distinctive cross-like pattern that no other order-8 group shares.
Click any of the five tables to expand it. The three abelian tables mirror across the diagonal; D4 and Q8 do not.
All five groups of order 8. Same number of elements, wildly different structure.
There are exactly five groups with 8 elements: three abelian (Z8, Z4×Z2, Z2³) and two non-abelian (D4, Q8). Click any table to expand it. Notice how the three abelian tables are symmetric while D4 and Q8 are not -- yet D4 and Q8 have strikingly different patterns from each other.
Key insight: Order alone does not determine a group. These five groups have identical size but completely different algebraic structure, visible immediately in their color patterns. The number of groups of order n grows rapidly -- order 16 already has 14 groups!
Click a thumbnail to compare D5, A4, D6 and Z12, and switch between color blocks and labels. Z12 stays symmetric while A4 does not.
Click a thumbnail to compare:
As groups grow larger, their tables become mesmerizing mosaics. Compare D5 (pentagon symmetries, order 10) with A4 (tetrahedron rotations, order 12) and spot how non-abelian groups create asymmetric, intricate patterns -- while Z12 remains perfectly symmetric despite its size.
Key insight: A4 is especially remarkable: it's the smallest group that proves not every group has subgroups of every divisor order. A4 has order 12 but no subgroup of order 6, violating the converse of Lagrange's theorem.
Spot a mistake or have an idea?