$$\langle G\mid \{abc\mid a,b,c \in G, abc = 1\}\rangle$$ is a presentation of $G$.
Why did we specifically choose the relations to be of length 3?
(Why not something like $\{ab\mid a,b \in G, ab = 1\}$ or $\{abcd\mid a,b,c,d \in G,abcd = 1 \}$?)
(This example appears in https://www.math.ucla.edu/~azhou/notes/1/210ABC/notes.pdf, page 34)
I found the notation confusing at first, hence my earlier comment, but once I saw the context, now I see the point.
The authors mean that any group $G$ has a presentation in which the set of generators is the whole set $G$, and the relations correspond to all the products. That is, for all $a, b \in G$, you write a relation $a b c = 1$, where $c = (a b)^{-1}$.