Given that the third term of a geometric progression (GP) is $-108$ and the sixth term is $32$, we are to find the common ratio, the first term, and other related details.

Step 1: Find the Common Ratio (r)

Let's denote the first term by $a$ and the common ratio by $r$. The $n$th term of a geometric sequence can be expressed as:

$T_n = ar^{n-1}$

For the third term ($T_3$):

$T_3 = ar^{3-1} = ar^2 = -108$

For the sixth term ($T_6$):

$T_6 = ar^{6-1} = ar^5 = 32$

We now have two equations:

$ar^2 = -108 \quad \text{(Equation 1)}$ $ar^5 = 32 \quad \text{(Equation 2)}$

Step 2: Solve for the Common Ratio (r)

Divide Equation 2 by Equation 1:

$\frac{ar^5}{ar^2} = \frac{32}{-108}$

Simplifying:

$r^3 = \frac{32}{-108} = \frac{32}{-108} = -\frac{8}{27}$

Thus,

$r = \sqrt[3]{-\frac{8}{27}} = -\frac{2}{3}$

So, the common ratio $r$ is $\boxed{-\frac{2}{3}}$.

Step 3: Find the First Term (a)

Using the value of $r$ in Equation 1:

$ar^2 = -108$

Substitute $r = -\frac{2}{3}$:

$a \left(-\frac{2}{3}\right)^2 = -108$

$a \cdot \frac{4}{9} = -108$

Multiply both sides by $\frac{9}{4}$:

$a = -108 \times \frac{9}{4} = -27 \times 9 = -243$

So, the first term $a$ is $\boxed{-243}$.

Step 4: Write the Geometric Sequence

With $a = -243$ and $r = -\frac{2}{3}$, the first few terms of the geometric sequence are:

1. First term: $T_1 = -243$
2. Second term: $T_2 = -243 \times \left(-\frac{2}{3}\right) = 162$
3. Third term: $T_3 = 162 \times \left(-\frac{2}{3}\right) = -108$
4. Fourth term: $T_4 = -108 \times \left(-\frac{2}{3}\right) = 72$
5. Fifth term: $T_5 = 72 \times \left(-\frac{2}{3}\right) = -48$
6. Sixth term: $T_6 = -48 \times \left(-\frac{2}{3}\right) = 32$

So, the geometric sequence is:

$\boxed{-243, 162, -108, 72, -48, 32, \dots}$

Step 5: Find the 10th Term

To find the 10th term ($T_{10}$):

$T_{10} = ar^{10-1} = ar^9$

Substitute $a = -243$ and $r = -\frac{2}{3}$:

$T_{10} = -243 \times \left(-\frac{2}{3}\right)^9$

Since $\left(-\frac{2}{3}\right)^9$ is negative, we get:

$T_{10} = -243 \times -\frac{512}{19683} = \frac{243 \times 512}{19683} = \frac{124416}{19683} = \boxed{6.32} \quad \text{(approx)}$

Would you like any further details or have any questions? Here are some related questions:

1. What is the general formula for the $n$th term of a geometric sequence?
2. How do you find the sum of the first $n$ terms in a geometric series?
3. How would the sequence change if the common ratio was a positive number?
4. What happens if the common ratio is a fraction greater than 1?
5. How can you determine if a sequence is geometric based on the terms provided?

Tip: When working with geometric sequences, it's important to consider whether the common ratio is positive or negative, as this will affect the sign of the terms in the sequence.