When the parents of a child live apart, the only person who has the automatic right to claim the child as a dependent is the parent or close relative (aunt or uncle, grandparent or sibling) who has physical custody of the child more than half the year -- the person whose home the child sleeps in more than half the nights of the year.
If your girlfriend, her sister, and your child all live together, then the child's aunt can claim the child as a dependent as long as the child lived in the aunt's home more than half the nights of the year AND the aunt earns more taxable income than the child's mother.
If your child does not live with your girlfriend's sister more than half the nights of the year, then she is not legally allowed to claim the child as a dependent on her tax return. But that doesn't mean that you can claim her. There may also be food stamp and medicaid fraud going on, but it's hard to tell without more information.
The dependent exemption can sometimes be transferred to a non-custodial parent such as yourself but it requires a signed form from the custodial parent, and would allow you to claim the child tax credit and the dependent exemption but NOT EIC, since that only ever stays with the custodial parent.
To know exactly who has what rights in this situation, we would need to know many more details about who lives with who and for how much of the time, and who pays for what amounts of support.
