Something like this?
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That’s the basic idea, yes, though I was thinking of a more angular design for the phoenix:
what you can infer of the federation based on its symbolism
Earth went through a bad patch, but has risen again and become a space-faring society. The fact that it's called the Neo-Terran Federation means that either all of those stars are colonies of Earth, or that Earth is the power and that the other stars are lesser members (thralls? conquered?).
I don't think these are meant to be the good guys, partly based on the phoenix styles you posted in your reply to the flag mock-up. "Federation" otherwise gives me good vibes, probably because of Star Trek's influence.
Actually a breakaway group after the original United Earth Commonwealth (or something) lost a war against an alien empire. A group of colonies and some sections of the Earth military were able to hold out and refused the surrender order, instead declaring independence.
They’re a bit more militarized and centralized than their “UN in Space” predecessor, following a bit after Heinlein type philosophies.
That works too! I went with my assumption because of the phoenix being between the blue (little blue dot) and black(ness of space).
The styles you were leaning towards were giving me somewhat fascist vibes, but it also makes sense for a militarised group. I suspect that as a rebel group they'll be more in the grey than straight good or bad, despite being (I assume) the good guys for the audience.