SymGen

Welcome to SymGen! This script generates multiplication tables for symmetry elements to test whether they form a complete group. Simply select the symmetry elements in your point group from those available below and then click “generate symmetry table”, or pick a point group which has already had its symmetry elements evaluated.

identity 1

inversion 1

reflection m[100] m[010] m[001] m[110] m[101] m[011] m[110] m[101] m[011]

proper rotation (order 2) 2[100] 2[010] 2[001] 2[110] 2[101] 2[011] 2[110] 2[111] 2[111] 2[111] 2[111] 2[111] 2[111] 2[111] 2[111]

proper rotation (order 3) 3[100] 3[010] 3[001] 3[110] 3[101] 3[011] 3[110] 3[111] 3[111] 3[111] 3[111] 3[111] 3[111] 3[111] 3[111] 3²[100] 3²[010] 3²[001] 3²[110] 3²[101] 3²[011] 3²[110] 3²[111] 3²[111] 3²[111] 3²[111] 3²[111] 3²[111] 3²[111] 3²[111]

proper rotation (order 4) 4[100] 4[010] 4[001] 4[110] 4[101] 4[011] 4[110] 4[111] 4[111] 4[111] 4[111] 4[111] 4[111] 4[111] 4[111] 4²[100] 4²[010] 4²[001] 4²[110] 4²[101] 4²[011] 4²[110] 4²[111] 4²[111] 4²[111] 4²[111] 4²[111] 4²[111] 4²[111] 4²[111] 4³[100] 4³[010] 4³[001] 4³[110] 4³[101] 4³[011] 4³[110] 4³[111] 4³[111] 4³[111] 4³[111] 4³[111] 4³[111] 4³[111] 4³[111]

proper rotation (order 6) 6[100] 6[010] 6[001] 6[110] 6[101] 6[011] 6[110] 6[111] 6[111] 6[111] 6[111] 6[111] 6[111] 6[111] 6[111] 6²[100] 6²[010] 6²[001] 6²[110] 6²[101] 6²[011] 6²[110] 6²[111] 6²[111] 6²[111] 6²[111] 6²[111] 6²[111] 6²[111] 6²[111] 6³[100] 6³[010] 6³[001] 6³[110] 6³[101] 6³[011] 6³[110] 6³[111] 6³[111] 6³[111] 6³[111] 6³[111] 6³[111] 6³[111] 6³[111] 6⁴[100] 6⁴[010] 6⁴[001] 6⁴[110] 6⁴[101] 6⁴[011] 6⁴[110] 6⁴[111] 6⁴[111] 6⁴[111] 6⁴[111] 6⁴[111] 6⁴[111] 6⁴[111] 6⁴[111] 6⁵[100] 6⁵[010] 6⁵[001] 6⁵[110] 6⁵[101] 6⁵[011] 6⁵[110] 6⁵[111] 6⁵[111] 6⁵[111] 6⁵[111] 6⁵[111] 6⁵[111] 6⁵[111] 6⁵[111]

rotoinversion (order 3) 3[100] 3²[100] 3[010] 3²[010] 3[001] 3²[001] 3[110] 3²[110] 3[101] 3²[101] 3[011] 3²[011] 3[110] 3²[110] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111] 3[111] 3²[111]

rotoinversion (order 4) 4[100] 4²[100] 4³[100] 4[010] 4²[010] 4³[010] 4[001] 4²[001] 4³[001] 4[110] 4²[110] 4³[110] 4[101] 4²[101] 4³[101] 4[011] 4²[011] 4³[011] 4[110] 4²[110] 4³[110] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111] 4[111] 4²[111] 4³[111]