What is the name of an order-16 non-abelian group $G$ satisfying the following properties?
Written in the group multiplication on the right: $$G=\langle a,b,c|a^2=b^2=c^2=1, ac = - ca, bc= -cb, ab = ba\rangle$$
This $G$ is generated by $a,b,c$, while it seems that it contains two kinds of dihedral group of order-8: $$ D_8=\langle ac, c| (ac)^2=-1, (ac)^4=1, (ac)c=-c(ac), ac = - ca\rangle $$ $$ D_8=\langle bc, c| (bc)^2=-1, (bc)^4=1, (bc)c=-c(bc), bc = - cb\rangle $$
The 16 group elements are: $$\{1,a,b,c,ab,bc,ac,abc,-1,-a,-b,-c,-ab,-bc,-ac,-abc\}$$
Hints:
Here are all the 14 kinds of order-16 non-abelian groups: https://groupprops.subwiki.org/wiki/Groups_of_order_16
I checked all of them above. The only group that seems to contain two dihedral group of order-8 $D_8$ as subgroups, is the $D_{16}$: https://groupprops.subwiki.org/wiki/Dihedral_group:D16 but the group I listed above does not contain an elementary abelian group $\mathbb{Z}$/8. So that was my puzzle.
Another possibility is $D_8 \times \mathbb{Z}/2$ https://groupprops.subwiki.org/wiki/Subgroup_structure_of_direct_product_of_D8_and_Z2, in that case, there should be four dihedral group of order-8 $D_8$ as subgroups. But I only identified two of them so far in this $G$.
Thank you in advance for the comments/answers!