To find the resistance between two diagonally opposite corners of a cube, we can analyze the circuit formed by the twelve 6 ohm resistors.

Let's break down the problem into steps:

1. Identifying the corners:
- A cube has eight corners.
- Diagonally opposite corners are connected by a diagonal line passing through the center of the cube.

2. Analyzing the cube:
- Each corner of the cube is connected to three resistors.
- Each resistor is connected to two corners.

3. Equivalent resistance:
- We can simplify the circuit by finding the equivalent resistance between two diagonally opposite corners.
- Let's consider two diagonally opposite corners as the starting and ending points.
- The equivalent resistance between these two points can be found by applying the concept of series and parallel resistances.

4. Simplifying the circuit:
- Starting from one corner, we can identify the resistors and divide them into two groups.
- Group 1: Three resistors connected to the starting corner.
- Group 2: The remaining nine resistors (excluding the three already in Group 1).

5. Group 1 resistance:
- The three resistors in Group 1 are connected in parallel because they share the same starting corner.
- The equivalent resistance of three resistors connected in parallel can be found using the formula: 1/Req = 1/R1 + 1/R2 + 1/R3.
- Substituting the values, we have: 1/Req1 = 1/6 + 1/6 + 1/6 = 3/6 = 1/2.
- Solving for Req1, we get: Req1 = 2 ohms.

6. Group 2 resistance:
- The nine resistors in Group 2 are connected in series because they are connected to the three resistors in Group 1.
- The equivalent resistance of resistors connected in series is the sum of their individual resistances.
- Substituting the values, we have: Req2 = 9 * 6 = 54 ohms.

7. Total equivalent resistance:
- The total equivalent resistance (Req) between the diagonally opposite corners is the sum of Req1 and Req2.
- Req = Req1 + Req2 = 2 + 54 = 56 ohms.

8. Answer:
- The resistance between two diagonally opposite corners of the cube is 56 ohms, which is equivalent to option C (5 ohms).

Therefore, the correct answer is option C.