The rule $(v_1,w_1)⋅(v_2,w_2)=(v_1+v_2,w_1+w_2+(v_1∧v_2))$ defines a group structure on the vector space $V⊕(V∧V)$ whenever $V$ is itself a vector space over some field $F$.
What is a more common name for this group?
- The group is nilpotent of class at most $2$, with commutator $[(v_1,w_1),(v_2,w_2)] = (0,2(v_1∧v_2))$.
- The group has exponent $\operatorname{char}(F)$ since $(v_1,w_1)^n = (v_1^n,w_1^n)$.
- When $\operatorname{char}(F) = 2$, this is just an elementary abelian $2$-group.
- When $\dim(V) = 1$, this is just the abelian group $V$, so has a faithful $2$-dimensional F-module.
- When $\dim(V) = 2$ and $\operatorname{char}(F)≠2$, this is a maximal unipotent subgroup of $\operatorname{GL}(3,F)$, so has a faithful $3$-dimensional $F$-module.
I don't recognize it when $\dim(V) = 3$.
When $\dim(V) = 3$ and $\operatorname{char}(F)≠2$, does the group have a faithful $F$-module of dimension independent of $F$?
Is this a maximal unipotent subgroup of some classical group or does it otherwise have a standard description as a matrix group?
I am primarily concerned with the case that $F$ is a finite prime field (and am hoping to do better than a permutation module of dimension $|F|^5$), but I assume many sorts of answers should be mostly independent of the field.
I don't have much hope for large $\dim(V)$, since the nilpotency class of the group remains at $2$, while maximal unipotent subgroups should increase their nilpotency class with their rank. Maybe $\dim(V)=3$ is small enough though.