- C1, identity
- C1, half turn around center
- C1, half turn around edge
- C1, half turn around corner
- C1, quarter turn around center
- C1, quarter turn around corner
- C1, orthogonal reflection through center
- C1, orthogonal reflection through corner
- C1, diagonal reflection
- C2_1, identity
- C2_1, quarter turn
- C2_1, orthogonal reflection
- C2_1, diagonal reflection
- C2_2, identity
- C2_2, orthogonal reflection
- C2_4, identity
- C2_4, quarter turn
- C2_4, orthogonal reflection
- C2_4, diagonal reflection
- C4_1, identity
- C4_1, orthogonal/diagonal reflection
- C4_4, identity
- C4_4, orthogonal/diagonal reflection
- D2_+1, identity
- D2_+1, half turn/other orthogonal reflection around/through center
- D2_+1, half turn/other orthogonal reflection around/through edge
- D2_+2, identity
- D2_+2, half turn/other orthogonal reflection around/through edge
- D2_+2, half turn/other orthogonal reflection around/through corner
- D2_x, identity
- D2_x, half turn/other diagonal reflection around/through center
- D2_x, half turn/other diagonal reflection around/through corner
- D4_+1, identity
- D4_+1, quarter turn/diagonal reflection
- D4_+2, identity
- D4_+4, identity
- D4_+4, quarter turn/diagonal reflection
- D4_x1, identity
- D4_x1, quarter turn/orthogonal reflection
- D4_x4, identity
- D4_x4, quarter turn/orthogonal reflection
- D8_1, identity
- D8_4, identity

- C1 => C1
- C1 => C2_1
- C1 => C2_2
- C1 => C2_4
- C1 => C4_1
- C1 => C4_4
- C1 => D2_+1
- C1 => D2_+2
- C1 => D2_x
- C2_1 => C2_1
- C2_1 => C4_1
- C2_1 => D4_+1
- C2_1 => D4_x1
- C2_2 => C2_2
- C2_2 => D4_+2
- C2_4 => C2_4
- C2_4 => C4_4
- C2_4 => D4_+1
- C2_4 => D4_x1
- C4_1 => C4_1
- C4_1 => D8_1
- C4_4 => C4_4
- C4_4 => D8_4
- D2_+1 => D2_+1
- D2_+1 => D4_+1
- D2_+1 => D4_+2
- D2_+2 => D2_+2
- D2_+2 => D4_+2
- D2_+2 => D4_+4
- D2_x => D2_x
- D2_x => D4_x1
- D2_x => D4_x4
- D4_+1 => D4_+1
- D4_+1 => D8_1
- D4_+2 => D4_+2
- D4_+4 => D4_+4
- D4_+4 => D8_4
- D4_x1 => D4_x1
- D4_x1 => D8_1
- D4_x4 => D4_x4
- D4_x4 => D8_4
- D8_1 => D8_1
- D8_4 => D8_4

Any ideas? I was thinking about a notation like D2_+1_h2, which gives the initial symmetry and has the the operation used tacked on at the end.