What is the name of this non-abelian group $G$?

Question

What is the name of an order-16 non-abelian group $G$ satisfying the following properties?

Written in the group multiplication on the right: $$G=\langle a,b,c|a^2=-1, b^2=c^2=1, ac = - ca, bc= -cb, ab =- ba, (-1)^2=1\rangle.$$

EDIT: The $-1$ is just a group element such that $(-1)^2=+1$.

This $G$ is generated by $a,b,c$, while it seems that it contains many dihedral groups of order-8: $D_8$.

The 16 group elements are: $$G=\{1,a,b,c,ab,bc,ac,abc,-1,-a,-b,-c,-ab,-bc,-ac,-abc\}$$

Hints:

1. Here are all the 14 kinds of order-16 non-abelian groups: https://groupprops.subwiki.org/wiki/Groups_of_order_16

2. I am not sure this $G$ contains any quaternion subgroups.

Answer 1

Ok, here is a precise way to write the presentation you want:

$$\langle a,b,c \mid a^4, b^2, c^2, [a^2, b], [a^2, c], ac = a^2ca, bc = a^2cb, ab = a^2ba \rangle$$

Here I am using $a^2$ to play the role of $-1$. Now I'll just punch this in to GAP:

gap> f := FreeGroup("a","b","c");;
gap> AssignGeneratorVariables(f);
#I  Assigned the global variables [ a, b, c ]
gap> r := ParseRelators([a,b,c],"a4,b2,c2,[a2,b],[a2,c],ac=a2ca,bc=a2cb,ab=a2ba");;
gap> StructureDescription(f/r);
"C2 x D8"

So, this group is isomorphic to $(\mathbb{Z}/2) \times D_8$.