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You can use whatever note, but we can use C. Unfortunately, I don't know what the root position would be in this case, so I will lay them out as positions. I used intervals, too.

Position 1: Root, major third,  major second, major third.



Position 2: Root, major second, major third,  major second.



Position 3: Root, major third,  major second, major third.



Position 4. Root, major second, major third,  major second.

I thought it could be some kind of augmented chord, as it has that augmented quality, but if it were an augmented seventh, wouldn't it be something like
Root, major third, major third, major second?

As is so often the case in music, a label depends on how something is functioning in context. There are several possibilities for this chord, and they can resolve variously to (at least) chords on B, F, E, or B♭. Prepare for a bit of an onslaught!

1. A French Augmented-Sixth Chord in E

Technically speaking, your listing of Root, major third, major second, major third actually results in C E F♯ A♯, because the major second above E is F♯ and not G♭. As such, you're looking at a French augmented-sixth chord. These chords tend to move to V (here, B), and so this chord is in the key of E:

2. Altered V7 in F

But if these pitches must be C E G♭ B♭, then it's an altered V7 in F. The V7 chord in F major is C E G B♭. But around the middle of the 19th century, composers started making adjustments to the members of the V7 chord; typically, these adjustments allowed for a smoother resolution to I. One common adjustment was to alter the chordal fifth, either by raising it or lowering it a half step. In this case, we can view this chord as a V7 with a lowered chordal fifth, or V7♭5. In so doing, the G♭ resolves smoothly by half step to F (compare that to what is normally a whole step from G♮ to F).

Those two are the most likely explanations, especially if you consider C to be the bass. But there are a few other options:

3. A French Augmented-Sixth Chord in B♭

One of the fascinating things about the French augmented-sixth is that it's symmetrical: it's two major thirds separated by tritone. Using your given pitches (and not the enharmonic ones in Explanation 1), we have an augmented-sixth chord not in E, but in B♭! But this requires G♭ to be in the bass. Once again, this chord will resolve to the dominant (V), which in B♭ will be F.

4. An Altered V7 in B

If we combine the enharmonic spelling of Explanation 1 with the harmonic function in Explanation 2, this can be viewed as a V7♭5 in B:

5. A "FrV4/3" in F

We can mix these altered dominant and augmented-sixth explanations to create yet more: one is an altered dominant (much like Explanations 2 and 4) that has G♭ in the bass (like Explanations 3 and 4).

The "French V4/3" is a V7 chord (in F, C is the root) with a lowered chordal fifth (G♭) in second inversion (this G♭ is the bass). A normal V7 in second inversion is labeled a V43, but since this lowered chordal fifth makes it enharmonic to a French augmented-sixth chord, we call it a FrV4/3.

The chord still resolves to F, but now it's understood that F is tonic (like in Explanation 2), not dominant (like in Explanations 3). The resolution is the same as in Explanation 3.

6. A "FrV4/3" in B

Lastly, the harmonic function of Explanation 5 meets the enharmonic spelling of Explanations 1 and 4: a FrV4/3 in B. This looks and sounds the same as Explanation 1, but the B chord is understood as tonic instead of dominant.

If you write C7b5 a pianist or guitarist will play the right notes. Or Gb(F#)7b5.

If you want a functional description, you'll need to show it in context, HAVING a function. There's a couple of tritones in there, either of which could power a resolution in two different directions!

This is a French Augmented 6th chord (Fr⁶).
Depending on the inversion used, the Minor 7th interval (C - B♭) tends to expand to an octave, while the inner voices resolve to the desired quality (ie. B Major).

Building off several of the existing answers, but putting it explicitly into a jazz context...

C#m7   C♮7b5    B6 

iim7   bII7b5   I6

...the second chord is the C E Gb Bb and would be a tritone substitution for the normal V7 in the common jazz progression iim7 V7 I6.

The enharmonic respelling of C E F♯ A♯ a French Augmented Sixth chord (in Richard's first example) would be the commonest functional use in 'classical' music.

Because the various spellings can be confusing, it's worth noting a French Augmented Sixth chord is an alteration of the more basic progression iiø43 V. In E minor that would be F# half diminished in second inversion, with the A tone raised. In terms of the E minor scale the 4th scale degree is altered by being raised with a sharp.

• start with iiø
  


• put it in second inversion


• raise the A with a sharp and resolve to V
  

It may take some time to wrap your head around the idea of augmented sixth chords, but I think it helps demystify them by recognizing their basic functions as inversions and alterations of either iiø7 or iv7 resolving to V in a minor key.

In addition to the ones already suggested, my first-choice instinctive interpretation was:

F# dominant that wants to resolve to a B tonic chord like Bm6. It's either missing the bass, or has a tritone substitution.

or:

D dominant that wants to resolve to a G tonic chord like G6. Try it, sounds very nice! D or G# bass note fits in very easily, but it doesn't necessarily need any added bass note.

or:

Ab dominant that wants to resolve to a Db tonic chord like Db6. (Thanks Richard)