Finite groups · Group

$\mathrm{Q}_{16}$ Generalized quaternion group of order sixteen

A generalized quaternion group with four real singlets, one real doublet and two quaternionic doublets.

Compare this group

Reference coverageFinite referenceNonabelian

Definition

The group with a⁸ = 1, x² = a⁴ and xax⁻¹ = a⁻¹.

Definition sources

Order: $16$
Center: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Order	$16$	Sources
Center	$\mathrm{C}_{2}$	Sources
Abelian	No	Sources
Simple	No	Sources
Ordinary complex irreducibles	1, 1, 1, 1, 2, 2, 2	Sources
GAP SmallGroups ID	SmallGroup(16,9)	Sources

Representation scope

The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities. [sources]

Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
ρ1	1	Real
ρ2	1	Real
ρ3	1	Real
ρ4	1	Real
ρ5	2	Real
ρ6	2	Quaternionic
ρ7	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.

Names and aliases

QuaternionGroup(16) · Dic4 · Dic(4)

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

Related objects and maps

$\mathrm{Dic}_{n}$
Dicyclic groups
Family
$\mathrm{D}_{8}$
Dihedral group of order eight
Q₁₆ / C₂ ≅ D₈

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

The group with a⁸ = 1, x² = a⁴ and xax⁻¹ = a⁻¹.