Specific heat for tetraatomic gas molecules
The heat capacity comprises contributions from translational motion (3 terms for each of $x, y, z$) plus (whole body) rotational motion ( 3 axis directions). Each motion accounts for $(1/2)R$ by equipartition theorem making the $(3/2)R + (3/2)R$ above. 

Then there is a contribution from the fact that the molecule is vibrating. Non-linear molecules have $3N-6=6$ vibrational modes. Each vibration has $(1/2)R$ contribution from its potential energy and the same from its kinetic energy which in total is the $6R$ in your question.