Center of unitriangular matrix group:UT(3,p)

Definition
Let $$p$$ be an odd prime (The case $$p = 2$$, where we get center of dihedral group:D8, is somewhat different, though many things are similar).

The group $$G = U(3,p)$$ is the group of unipotent upper-triangular matrices over the prime field of $$p$$ elements, and is also the unique (up to isomorphism) non-abelian group of order $$p^3$$ and exponent $$p$$. It is defined by the presentation:

$$G := \langle g,h,k \mid g^p = h^p = k^p = e, gh = hg, gk = kg, khk^{-1} = g^{-1}h \rangle$$.

In the matrix description, each matrix $$(a_{ij})$$ can be described by the three entries $$a_{12}, a_{13}, a_{23}$$. The matrix looks like:

$$\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23}\\ 0 & 0 & 1\end{pmatrix}$$

The multiplication of matrices $$A = (a_{ij})$$ and $$B = (b_{ij})$$ gives the matrix $$C = (c_{ij})$$ where:


 * $$c_{12} = a_{12} + b_{12}$$
 * $$c_{13} = a_{13} + b_{13} + a_{12}b_{23}$$
 * $$c_{23} = a_{23} + b_{23}$$

With the matrix description, we can set $$g$$ as the matrix with $$a_{13} = 1$$ and the other two entries zero, $$h$$ as the matrix with $$a_{12} = 1$$ and the other two entries zero, and $$k$$ as the matrix with $$a_{23} = 1$$ and the other two entries zero.

Now define the subgroup:

$$H := \{ \begin{pmatrix} 1 & 0 & a_{13} \\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix} : a_{13} \in \mathbb{F}_p \}$$

In other words, this is the subgroup given by $$a_{12} = a_{23} = 0$$.

Cosets
The subgroup has $$p^2$$ cosets each of size $$p$$, where each coset is parametrized by fixed values of a_{12} and $$a_{23}$$. In other words, for any fixed $$a_{12} \in \mathbb{F}_p$$ and $$a_{23} \in \mathbb{F}_p$$, the corresponding coset of $$H$$ in $$G$$ is the set of all matrices with those values of $$a_{12}$$ and $$a_{23}$$, and with $$a_{13}$$ varying over all of $$\mathbb{F}_p$$.

Complements
The subgroup has no permutable complements and also no lattice complements.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many subgroup-defining functions on the whole group. Some of these are given below.