Finite groups · Group

$\mathrm{D}_{8}$ Dihedral group of order eight

The full symmetry group of a square. This site uses D₈ to mean order eight; many references call it D₄.

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Definition

⟨r,s | r⁴ = s² = 1, srs = r⁻¹⟩

Definition sources

Order: $8$
Center: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Order	$8$	Sources
Center	$\mathrm{C}_{2}$	Sources
Abelian	No	Sources
Simple finite group	No	Sources
Ordinary complex irrep dimensions	1, 1, 1, 1, 2	Sources

Representations

Four one-dimensional irreps and one real-type doublet. Its irrep dimensions match Q₈, but the doublet has a different reality type. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
$1$	1	Real
1a	1	Real
1b	1	Real
1c	1	Real
$2$	2	Real

Complete dimension and reality summary for ordinary complex irreps. This does not supply explicit matrices, every character value, or modular representations. Labels follow the conventions stated in this record.

In physics

Point-group actions, selection rules, and comparisons of groups with similar character data. [sources]

Names and aliases

D4 · Square symmetries

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathrm{Q}_{8}$
Quaternion group
Related group
$\mathrm{S}_{3}$
Symmetric group on three letters
Related group
$\mathrm{D}_{2n}$
Finite dihedral groups
Dihedral family; order eight

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

⟨r,s | r⁴ = s² = 1, srs = r⁻¹⟩