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