Multiplication table for the binary tetrahedral group

The elements are denoted by the symbols derived in the Quaternionic representation of the binary tetrahedral group.

NOTE: since each element has a negative, the table should be four times as large; but $-1$ commutes with everything, so the missing products are easy to retrieve.

${\large \begin{array} {c ||c | c | c | c | c | c | c | c | c | c | c | c ||} & 1 & {\bf i} & {\bf j} & {\bf k} & a & a^2& b & b^2& c & c^2& d & d^2\\\hline\hline 1 & 1 & {\bf i} & {\bf j} & {\bf k} & a & a^2& b & b^2& c &c^2 & d & d^2\\\hline {\bf i} & {\bf i} & -1& {\bf k} & -{\bf j} & d &-c^2& c &-d^2& -b&a^2 & -\ a & b^2\\\hline {\bf j} & {\bf j} & -{\bf k}& -1& {\bf i} & b &-d^2& -a & c^2& d &-b^2& \ -c & a^2\\\hline {\bf k} & {\bf k} & {\bf j} & -{\bf i}& -1 & c &-b^2& -d & a^2& -a &d^2 & \ b &-c^2\\\hline a & a & c & d & b & a^2& -1 &-c^2& {\bf j} &-d^2& {\bf k} &-b^2& {\bf i} \\\hline a^2&a^2 &-d^2&-b^2&-c^2& -1& -a & - {\bf k}& d & -{\bf i} & b & -{\bf j} & c\ \\\hline b & b & d & -c & -a &-d^2& -{\bf j} & b^2& -1 &-a^2& {\bf i} &-c^2&-{\bf k}\ \\\hline b^2& b^2&-c^2& a^2& d^2& {\bf k} & c & -1 & -b & {\bf j} & d & -{\bf i} & a\ \\\hline c & c & -a & b & -d &-b^2& -{\bf k} &-d^2& -{\bf i} & c^2& -1 &-a^2&{\bf j} \ \\\hline c^2&c^2 & b^2&-d^2& a^2& {\bf i} & d & -{\bf j} & a & -1 & -c & {\bf k} &b \ \\\hline d & d & -b & -a & c &-c^2& -{\bf i} &-a^2& {\bf k} &-b^2& -{\bf j} & d^2&-1\ \\\hline d^2& d^2& a^2& c^2&-b^2& {\bf j} & b & {\bf i} & c & -{\bf k} & a & -1 &-d\ \\\hline\hline \end{array}}$

Reset using MathJax, August 21 2024