I am studying the computer algebra system GAP to do some calculations about Clifford group, which is defined (cf. Lawson and Michelsohn, Spin Geometry, Princeton 1989) as followings Definition. Let's denote this group by $F_n$, in accordance with the number of generators $e_1, e_2, \ldots, e_n$. The group can be given $F_n = \, <\;e_1, e_2, \ldots, e_n \,:\; e_j^2 = -1, e_j e_k = - e_k e_j \quad \text{for } k \ne j; 1\leq j, k \leq n \;>.$
Thus we can calculate the order of this group, it is equal to $2^{n+1}$. Given a number $n$, for example, $n=3 \text{ or }4$, I want to identify these groups with the groups listed in the Small Group Library with precise ID number.
I want to know how to define this group $F_n$ in GAP, I tried like followings but failed Define $F_3$
My questions are
- How to modify the codes in my definition to get a correct one? I think the main problem is $e_j^2=-1$.
- Once we can define this group, how do we identify the group with the group in Small Group Library?
- Since GAP can handle group whose order is less than about 2000. For the Clifford group case, this means I can work up to $n=8$ when the group order is $2^9=512$. The Small Group Library does not have group of order $1024$, and group of order $2^{11}$ and beyond are also not handles. How to deal with those $n \geq 9$ situations?
Many thanks.