Can my ex-girlfriend file child support even though my name is not on the baby's birth certificate?

[Eric Brant]

Here is the back story:

Throughout the entire pregnancy, she has kept much of the information pertaining to the baby away from me. Several times she would talk to me civilly, and then suddenly block me (on Facebook); telling me that she wants nothing to do with me.
This morning she had the baby, and her friend keeps on harassing me, saying that I am a "dead-beat father". How could this be? I have never even had the opportunity to even visit the baby. My ex-girlfriend did not tell me when she went into labor. I had to find that out via a second Facebook of mine that I had to make.

This really bums me out, because I was looking forward to the moment of witnessing my first child being born, but because my ex allows her little friend to fill her head with condescending bs about me, she has totally betrayed me even though she promised that she would at least let me know when the baby was to arrive.Her friend has caused a lot of drama and dysfunction in my life that it has disabled any ability of making this situation better.

Just a few moments ago, my ex-girlfriend contacted me, telling me that she did [not] put my name on the birth certificate. In fact, she changed the baby's name to an entirely different name without my pre-knowledge or input. Honestly, I do not care about my ex-girlfriend, because our relationship with one another is miserable. She is emotionally abusive; I just do not want that in my life anymore. What I care about is this baby who will ultimately grow up without a father. Tthat be the fault of his selfish mother.

Sadly, there is question about whether or not I am the father, because there is no honest certainty that I was the only man in her life. She is very known by pretty much everyone for her promiscuous history. I want to take a DNA test, but she will not agree to it.

She will not agree to be taking a DNA test.
She did not put my name on the baby's birth certificate.

Am I allowed to request a DNA test, or does that privilege lie with the mother?