C2×C2×C2

C2×C2×C2 denotes the direct product of three copies of the cyclic group of order 2. In algebraic terms it is the elementary abelian 2-group of rank 3, commonly written as (Z/2Z)^3 or F2^3, the three-dimensional vector space over the field with two elements.

It has eight elements: the zero vector and all eight binary 3-tuples. The group operation is component-wise addition modulo 2. Every nonzero element has order 2, so the group is not cyclic since no element has order 8.

Subgroup structure corresponds to subspaces of the F2^3 vector space. There are seven subgroups of order 2, each generated by a nonzero vector, and seven subgroups of order 4, each forming a 2-dimensional subspace. This lattice reflects its interpretation as a vector space: subgroups are precisely the linear subspaces.

The automorphism group of C2×C2×C2 is GL(3,2), of order 168. This group acts transitively on the nonzero elements and on the one-dimensional subspaces, preserving the vector-space structure. GL(3,2) is isomorphic to PSL(2,7), linking the object to several areas in finite geometry and group theory.

In broader contexts, C2×C2×C2 appears as the translation subgroup of the affine space AG(3,2) and as a basic example of an elementary abelian p-group in group theory. It also serves as a simple model for binary linear algebra and has connections to combinatorial structures such as the Fano plane and related symmetry groups.

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