Keys: r rotate, s reflect, z undo. The dashed line is the fixed axis of s.
Current element
e
The identity: nothing moves.
- Order
- 1
- Vertices
- ()
Moves so far
none
Products are read right to left: in r²s the s acts first. Each new move goes on the left, and the rule sr = r⁻¹s brings any word to the form rᵏ or rᵏs.
All 8 elements
Rotations
Reflections
Click one to move there. Hover a reflection to see its axis.
The element e in each representation
A representation turns every symmetry into a matrix, so that doing two symmetries in a row multiplies their matrices. These are the irreducible ones; every other representation of D4 is built from them.
Each dial shows where its representation sends an upright flag. E1 turns with the polygon; E2 turns twice as far for each r, and so on. The dots are the places r can take the flag; a reflection flips the orange tick to the other side.
The dial shows where E sends an upright flag. It turns with the polygon. The dots are the places r can take the flag; a reflection flips the orange tick to the other side.
Character table
The character of a representation is the trace of its matrices. It only depends on the conjugacy class, so there is one column per class (the small number is the class size). The current element's column is lit; click a class to move to it.
Dimensions squared: , the order of the group.
The last row permutes the 4 vertices; its character counts the vertices an element leaves in place.
Tensor products
Product row ; dimension .
Multiply the two character rows class by class, then take the inner product ⟨χ, ψ⟩ with each irreducible row, where C runs over the classes. The results are the multiplicities.