Which primitive unit cell has unequal edge lengths ($$a \neq b \neq c$$) and all axial angles different from 90$$^{\circ}$$?

In the language of solid-state chemistry, every crystal system is recognised by specifying two sets of geometric parameters:

• the three edge lengths, written as $$a,\;b,\;c$$, and
• the three axial (inter-axial) angles, written as $$\alpha,\;\beta,\;\gamma$$.

When we compare any stated unit cell with the seven recognised crystal systems, we match these six parameters with the characteristic conditions that define each system. Before matching, let us recall the essential conditions for the four systems mentioned in the options.

Hexagonal system (Option A):
We have $$a=b\neq c$$ and the angles satisfy $$\alpha = \beta = 90^{\circ},\;\gamma = 120^{\circ}.$$ Clearly, here two edges are equal, so $$a=b$$, which does not agree with $$a\neq b\neq c$$.

Monoclinic system (Option B):
We have $$a\neq b\neq c$$ (so the edges are indeed unequal) but only one of the three axial angles is different from $$90^{\circ}$$. Specifically, $$\alpha = \gamma = 90^{\circ},\;\beta \neq 90^{\circ}.$$ Thus, in the monoclinic case, two angles remain right angles. This fails the requirement “all axial angles different from $$90^{\circ}$$.”

Triclinic system (Option C):
We have $$a\neq b\neq c$$ and simultaneously $$\alpha \neq \beta \neq \gamma \neq 90^{\circ}.$$ That is, every edge length is distinct and all three angles are individually oblique (none equals $$90^{\circ}$$). This matches the description in the question word for word.

Tetragonal system (Option D):
We have $$a=b\neq c$$ together with $$\alpha = \beta = \gamma = 90^{\circ}.$$ Here again two edges are equal and every angle is a right angle, so the condition fails in two ways.

Only the triclinic (also called anorthic) primitive unit cell fulfils the twin conditions $$a\neq b\neq c$$ and $$\alpha\neq\beta\neq\gamma\neq 90^{\circ}.$$

Hence, the correct answer is Option C.