Why can't I find this group anywhere online?

[davidbenari]

http://i.imgur.com/JgpJp03.png

I've done the Cayley table for the group above and can't find it in any of the group encyclopedias online. I can post it too if you want, but I'll tell you this:

It is a non abelian group of order 8 with two generators (a,g) such that a^4=Identity and g^4=identity. Plus, it only has three self inverses (apart from the identity).

No group seems to satisfy this. Anyone knows which group I'm talking about?

Should I post the Cayley table I've done too?edit:

And here is the Cayley table http://i.imgur.com/VKJr18F.jpg

 

[Orodruin]

If you do a little searching, you will find that there are two non-Abelian groups of order 8. The dihedral group $D_4$ and the quaternion group. I will let you figure out which one is the correct one.

 

[Orodruin]

Also, your Cayley table does not match the Cayley graph. For example, $b^2 = d$ and not the identity.

 

[davidbenari]

Hmm. I'm checking that $b^2=d$. For what its worth, I had already checked $D_4$ and the quaternion group and neither match my table. $D_4$ has 6 self inverses, the quaternion group has $2$ (I'm including the identity here).

Even if $b^2=d$ and my Cayley table is a wrong representation of the diagram, the Cayley table I've shown look quite legit and its making me crazy why I can't find it online.
By the way, and this might be a dumb question, but how did you figure out $b^2=d$?

 

[davidbenari]

I've figured out $b^2=d$ now. Just for the heck of it, as I said, the Cayley table looks legit so why shouldn't it be online? My opinion is that there is something wrong somewhere with my table. After all, it is a proven fact that there is only a certain quantity of groups of order 8, right?

 

[Orodruin]

Your Cayley table does not satisfy associativity. For example, $(ba)g = 1$ while $b(ag) = d$. Therefore, it does not describe the multiplication table of a group.

 

[Orodruin]

And yes, there are only five groups (up to isomorphisms) of order 8. They are the Abelian groups $C_8$, $C_4\times C_2$, and $C_2^3$ and the non-Abelian groups already mentioned.