Search the full group $G$ based on the partial list of its matrix representations

Suppose I have a list of matrices that may represent the partial list of the full group $G$. And here are the given set of 7 matrix elements.

$$e =\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ $$i =\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$ $$j =\begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$$ $$k =\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ $$\overline{e} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$$ $$\overline{i} =\begin{pmatrix} -1 & -1 \\ -1 & 1 \end{pmatrix}$$ $$\overline{j} =\begin{pmatrix} 1 & -1 \\ -1 & -1 \end{pmatrix}$$

How do I use Mathematica function to effectively

(1) find the full group $G$ based on the above lists?

(2) find the missing group elements in the $G$ not given in the above list? Thus, also determine the total number of group elements in the $G$?

(3) How to find a Multiplication Table in terms of the entry of these matrix representations? (for example, using FiniteGroupData["Quaternion", "MultiplicationTable"] )

(The above question is just a trial problem. In this case, I already know the answer: The missing entry is $\overline{k} =\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, and the full group is the order 8 qauternion.)

But how do we find an effective algorithm in Mathematica to help to solve similar problems?

Thanks!

(Edit: Finding the Multiplication Table is enough, no need for character table...)

• (1) and (2) can be done via a FixedPoint and Union scheme like FixedPoint[Union[Table[Map[el*#&, #], {el, #}]], els] or something. I use a similar tactic here to (inefficiently) enumerate all of the symmetry planes in a molecule. I only know (3) in the context of symmetry operations so can't help you there. This or this might help though. – b3m2a1 Apr 10 '18 at 1:49
• @ b3m2a1 +1, feel free to write an answer to clarify how to input these 7 matrices to generate the full group. I guess that as long as one can obtain the full set of group elements $\{g_a\}$, then one can compute the character table from the multiplications $g_a g_b$, then it is enough for me to compare with some mathematics group theory data book. – wonderich Apr 10 '18 at 2:40
• (Edit: Finding the Multiplication Table is enough, no need for character table...) – wonderich Apr 10 '18 at 3:19
• @ b3m2a1, I am not sure that I can read or digest this --" FixedPoint[Union[Table[Map[el*#&, #], {el, #}]], els]"... -- can you write an answer for it: ? – wonderich Apr 10 '18 at 4:08