To give an alternative answer, let us first recall the article "Construction of Locally Compact Near-Fields from p-Adic Division Algebras" by Detlef Groger:

Fix a prime element $\pi_F$ of $F$. Then $D$ is generated as a non-commutative $F$-algebra by an unramified extension $E/F$ of degree 2 and an element $\pi$ with $\pi^2=\pi_F$. we consider $\varpi$ for a (fixed) primitive $(p^2 -1)^{th}$ root of unity in $E$ and $ \varpi_F=\varpi^\frac{p^2 -1}{p-1} \in F$

Suppose $U_F,U_D$ denote the group of units in $F$ and $D$ with $U^1_F := 1 + P_F$ and $U^1_D := 1 + P_D$. Put $C = < \varpi, \pi >$ and $C_F=C \cap F^* = < \varpi_F, \pi_F >$. Then, $C$ is a complement of $U^1_D$ in the semidirect product $D^*=U^1_D \rtimes C$, whereas the product $F^*=U^1_F \times C_F$ is direct product. Therefore, $F^* U^1_D= U^1_D \times C_F$ and, $$\frac{D^*}{F^*(1+P_D)} \cong C/C_F \cong <\overline{\varpi}> \rtimes <\overline{\pi}> \cong Z_{p+1} \rtimes Z_2 .$$

To give an alternative answer, let us first recall the article "Construction of Locally Compact Near-Fields from $p$-Adic Division Algebras" by Detlef Groger:

Fix a prime element $\pi_F$ of $F$. Then $D$ is generated as a non-commutative $F$-algebra by an unramified extension $E/F$ of degree 2 and an element $\pi$ with $\pi^2=\pi_F$. we consider $\varpi$ for a (fixed) primitive $(p^2 -1)^\text{th}$ root of unity in $E$ and $ \varpi_F=\varpi^\frac{p^2 -1}{p-1} \in F$.

Suppose $U_F$, $U_D$ denote the groups of units in $F$ and $D$ with $U^1_F := 1 + P_F$ and $U^1_D := 1 + P_D$. Put $C = \langle \varpi, \pi \rangle$ and $C_F=C \cap F^* = \langle \varpi_F, \pi_F \rangle$. Then, $C$ is a complement of $U^1_D$ in the semidirect product $D^*=U^1_D \rtimes C$, whereas the product $F^*=U^1_F \times C_F$ is direct product. Therefore, $F^* U^1_D= U^1_D \times C_F$ and, $$\frac{D^*}{F^*(1+P_D)} \cong C/C_F \cong \langle\overline{\varpi}\rangle \rtimes \langle\overline{\pi}\rangle \cong Z_{p+1} \rtimes Z_2 .$$