Problems with every day objects (repost): Baby Building Blocks

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Cube Net depicted above


Today, while sitting in the doctors office waiting, I saw four identical cubes with 3 white and 3 red faces. I couldn't help myself so I began to play with them... And by play with them, I mean that I began to make a math problem about how many arrangements I could make with them. So here it is:

When lined up side by side and only looking at the top face of each cube. How many different arrangements of the identical cubes can I make such that there is no \(180^{\circ}\) rotational symmetry along the x,y, or z axis or around the origin.

Assume:

The origin is the center of the 4x1x1 rectangular prison formed by aligning the four blocks.

If the letters have the same orientation and order as another combination but has a different color order, it's considered different and does not have reflectional symmetry.

Hint: if the top of the four blocks are in this specific order \(\boxed{\lfloor}\boxed{\lceil}\boxed{\rfloor} \boxed{\rceil}\) it is rotationally symmetric Around the y axis.

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