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
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.