# Projective representation theory of dihedral group:D8

This article gives specific information, namely, projective representation theory, about a particular group, namely: dihedral group:D8.

View projective representation theory of particular groups | View other specific information about dihedral group:D8

This article describes the projective linear representations of dihedral group:D8.

The Schur multiplier of dihedral group:D8 is cyclic group:Z2 (see group cohomology of dihedral group:D8 and second cohomology group for trivial group action of D8 on Z2). There are three possible choices for the Schur covering group: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. We pick dihedral group:D16 for this discussion. The projective representations of dihedral group:D8 all arise from linear representations of dihedral group:D16 (see linear representation theory of dihedral group:D16).

Representation | Degree | Corresponding element of Schur multiplier, which is cyclic group:Z2 | Number of ordinary representations of dihedral group:D16 which give rise to it | List of these (see linear representation theory of dihedral group:D16) |
---|---|---|---|---|

trivial | 1 | trivial (identity element) | 4 | all the one-dimensional representations |

two-dimensional irreducible linear | 2 | trivial (identity element) | 1 | this representation has kernel the center of dihedral group:D16 and descends to the faithful irreducible representation of dihedral group:D8. |

two-dimensional irreducible projective, not linear on |
2 | nontrivial (non-identity element) | 2 | either of the two choices for the faithful irreducible representation of dihedral group:D16 (they are projectively equivalent) |