Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

Question

If we have a group presentation $G=\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$, how we will get the following values:

1. The order of the group.

2. The elements of the group written in terms of the generators

Answer 1

This group is the same as $\langle a,b,c : a^3 = c, b^3 = c, c^3 = 1, ba=abc, ca=ac, cb=bc \rangle$. This presentation is a reduced, confluent polycyclic presentation, so $G$ has order $27$ (the product of the relative orders of the generators) and its elements are exactly the products $a^i b^j c^k$ for $0 \leq i,j,k < 3$.

You can check in GAP that this group is SmallGroup(27,4), the extraspecial group of exponent 9 and order 27.

gap> f:=FreeGroup("a","b");;
gap> IdGroup(f/ParseRelators(f,"a^9=1, b^3=a^3, [a,b]=a^3"));
[ 27, 4 ]