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We should already have the naming for string 6, but let's double check. We've chosen the tone names for the sharps/flats which correspond to the Major chords [when we modify this form to a minor, some of the root names change]. Once we get into a shape, it can be shifted to every position. 12 things, for knowing 1 thing, is good mileage. And, 3, 4, & 5 string versions are available; some fretted with individual fingers. Sometimes this full version [6 string version] can be called a power chord, whether fully fingered as shown, or fretting just the bottom tones [the 6 and 5 strings - and the 4 string can be added as well]. Yet, the term power chord has come to mean just the R5 types, in common usage.
First, we move the shape, as is, fret by fret, leaving the 6, 2, & 1 strings open. At each fret, let's make a thumbs up or down for the way the chord rings.
For thumbs up, the chord will need to resonate in a way that is somewhat consonant. For thumbs down, it will sound like a scary movie chord. Thumbs down doesn't mean unusable, just dissonant. The scary movie chords, when arpeggiated, can sound pretty cool. For our purposes here, we are thinking about strumming.
These chords are labeled by the tones which are being fretted. Obviously, with the 6, 2, and 1 strings ringing, some of the chord names will be different. The exception is the E chord. We aren't overly concerned with their technical names; we are just seeking out cool sounds. Many of them actually work for what they are named. Example: G is actually a G6/E, but will work fine when we see a G. We've chosen flat names for the sharp/flat. Most of the sharp/flat Major triads have flat names [F# and Gb are both].
If the system is closed [no open strings/only the 5, 4, & 3 fretted], the chord name is the name of the chord, & the chord will sound fine everywhere.
On some, like the F, we can control the low E with our fretting hand thumb. By doing this, we can not let it ring as loud, or mute it out completely.
You know this chord. Fret it with confidence. Make sure that the 2 & 1 strings are ringing clearly. The 6 string will ring, but we can control how much it rings by using our fretting hand thumb. The six string will factor into thumbs up or down, but by controlling it [dampening or muting], some chords get through the process on the side of thumbs up.
The grid shows the normal fingering. E could also be fretted with the 3, 4, & 2 fingers, which is not uncommon [this is a 'barless' E form].