Finite groups · Group

$\mathrm{Q}_{8}$ Quaternion group

The eight unit quaternions {±1, ±i, ±j, ±k}, naturally a finite subgroup of SU(2).

Compare this group

Executable coverageSpin coverMcKay

Definition

i² = j² = k² = ijk = −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 real-type one-dimensional irreps and one quaternionic-type doublet. A real-valued character does not imply real 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	Quaternionic

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

Projective symmetries, quaternionic representations, and the affine D₄ McKay graph. [sources]

Names and aliases

Quaternion group of order 8

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

Related objects and maps

$\mathrm{SU}(2)$
Special unitary group
Related group
$\mathrm{D}_{8}$
Dihedral group of order eight
Related group
$\mathrm{Dic}_{n}$
Dicyclic groups
Dic₂

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

i² = j² = k² = ijk = −1