# Dihedral group of order 8

The dihedral group of order 8, usually written D₄ or D₈, is the group of the eight symmetries of a square: four rotations and four reflections, combined by composition. It is one of the two non-abelian groups among the five groups of order 8 and is sometimes called the octic group.<sup>[1](https://mathworld.wolfram.com/DihedralGroupD4.html)</sup><sup> • </sup><sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup>

| Key fact | Detail |
|---|---|
| Order and structure | 8 elements; non-abelian; the symmetry group of the square<sup>[1](https://mathworld.wolfram.com/DihedralGroupD4.html)</sup> |
| Presentation | ⟨ s, t ; s⁴ = t² = e ; ts = s³t ⟩, with s a quarter-turn rotation and t a reflection<sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup> |
| Elements | 1, s, s², s³, t, ts, ts², ts³; four rotations and four reflections<sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup> |
| Classification | One of five groups of order 8, alongside C₈, C₄ × C₂, C₂ × C₂ × C₂, and the quaternion group Q₈<sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup> |
| GAP identifier | SmallGroup(8,3); also labeled He₂ = AΣL₁(𝔽₄) = 2₊^(1+2)<sup>[3](https://people.maths.bris.ac.uk/~matyd/GroupNames/)</sup> |
| Center | ⟨r²⟩, the subgroup {e, r²} of order 2, since Z(Dₙ) = ⟨r^(n/2)⟩ for even n<sup>[4](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> |
| Automorphism group | Isomorphic to D₈ itself<sup>[5](https://www.math.colostate.edu/~jwilson/math/8/D8.html)</sup> |

## Elements and geometric meaning

Number the vertices of a square 1, 2, 3, 4. The four rotations are the quarter-turn r = (1234), the half-turn r² = (13)(24), the three-quarter-turn r³ = (1432), and the identity r⁴ = (1). The four reflections complete the set: two across the diagonals, and two across the lines through midpoints of opposite sides, giving the remaining elements rs and r³s as the double transpositions (12)(34) and (14)(23).<sup>[6](https://web.math.utk.edu/~dcartwr1/iaawa/section-dihedral-groups.html)</sup>

The group is generated by a rotation r of order 4 and a reflection s of order 2, subject to the relation srs⁻¹ = r⁻¹, which says that reflecting reverses the direction of rotation. Every element then reduces to the normal form rⁱ or rⁱs with 0 ≤ i ≤ 3, giving exactly eight elements: e, r, r², r³, s, rs, r²s, r³s.<sup>[7](https://johndcobb.github.io/intro-abstract-algebra-notes/lectures/01-introduction-to-groups)</sup>

The relation srs⁻¹ = r⁻¹ is also what makes the group non-abelian for n ≥ 3: if sr = rs held, cancellation would force r² = e, contradicting the fact that r has order 4.<sup>[7](https://johndcobb.github.io/intro-abstract-algebra-notes/lectures/01-introduction-to-groups)</sup>

## Presentation

A standard presentation is

\[ D_4 = \langle a, b \mid a^4 = 1,\; b^2 = 1,\; ba = a^{-1}b \rangle, \]

where a is a rotation and b is a flip.<sup>[8](https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/Makeup1-ModernAlgebra-S09.pdf)</sup> An equivalent form uses the relations a⁴ = b² = (ab)² = 1; this presentation arises in the classification of the groups of order 8 as the case where an element b outside the cyclic subgroup ⟨a⟩ satisfies b² = 1 and ba = a³b.<sup>[9](https://crypto.stanford.edu/pbc/notes/group/ordereight.html)</sup>

The group also has a two-dimensional matrix representation over the complex numbers, with the rotation generator s represented by the diagonal matrix diag(i, −i) and the reflection generator t by an antidiagonal matrix, both 2 × 2.<sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup>

## Place among the groups of order 8

There are five groups of order 8 up to isomorphism: the three abelian groups C₈, C₄ × C₂, and C₂ × C₂ × C₂, and the two non-abelian groups D₄ and Q₈.<sup>[2](https://mathweb.ucsd.edu/~atparris/small_groups.html)</sup> The classification argument distinguishes them by the behavior of an element b outside a cyclic subgroup of order 4: b² = 1 with ba = a³b gives the dihedral group, while b² = a² with ba = a³b gives the quaternion group.<sup>[9](https://crypto.stanford.edu/pbc/notes/group/ordereight.html)</sup>

**D₄ versus Q₈.** The two non-abelian groups of order 8 are not isomorphic, and the standard test compares the numbers of elements satisfying x² = e: D₄ has more involutions than Q₈, which has only one.<sup>[7](https://johndcobb.github.io/intro-abstract-algebra-notes/lectures/01-introduction-to-groups)</sup> Yet the pair is famously close: D₄ and Q₈ are non-isomorphic groups that nevertheless share the same character table.<sup>[10](https://mathworld.wolfram.com/CharacterTable.html)</sup> In GAP's SmallGroups library D₄ carries the identifier (8,3) and the alternative names He₂, AΣL₁(𝔽₄), and the extraspecial group 2₊^(1+2), while Q₈ is (8,4).<sup>[3](https://people.maths.bris.ac.uk/~matyd/GroupNames/)</sup>

## Subgroups and normal structure

The subgroup structure is small enough to list completely. The rotation subgroup ⟨r⟩ of order 4 is cyclic, and the center is ⟨r²⟩ = {e, r²}; in general Z(Dₙ) = ⟨r^(n/2)⟩ when n is even, which for n = 4 gives this order-2 center.<sup>[4](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> The composition factors of the group are three copies of C₂, reflecting the chain of normal subgroups from the whole group down through ⟨r²⟩ to the identity.<sup>[5](https://www.math.colostate.edu/~jwilson/math/8/D8.html)</sup>

Conjugation by a reflection sends each rotation rⁱ to r⁻ⁱ, so the rotation subgroup ⟨r⟩ is normal, and its normalizer is the whole group; by contrast the normalizer of a reflection subgroup ⟨s⟩ is ⟨s, r^(n/2)⟩ when n is even, which for D₄ is the [Klein four-group](https://www.edgepedia.org/klein-four-group) ⟨s, r²⟩.<sup>[4](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)</sup> The automorphism group of D₄ is isomorphic to D₄ itself.<sup>[5](https://www.math.colostate.edu/~jwilson/math/8/D8.html)</sup>

## Representations

The natural action on the square's four vertices gives a 4-dimensional permutation representation, in which each group element permutes the vertices and is represented by a 4 × 4 permutation matrix.<sup>[11](https://carstenjaeschke.com/essays/representation-theory.html)</sup> The standard 2-dimensional real representation, given by the rotation and reflection matrices acting on the plane, is irreducible over ℝ: no line through the origin is invariant under all of D₄, because a 90° rotation maps each candidate line to a different one.<sup>[11](https://carstenjaeschke.com/essays/representation-theory.html)</sup>

The group's cycle index, which records how its elements permute the four vertices, is

\[ Z(D_4) = \tfrac{1}{8}x_1^4 + \tfrac{1}{4}x_2 x_1^2 + \tfrac{3}{8}x_2^2 + \tfrac{1}{4}x_4. \]<sup>[1](https://mathworld.wolfram.com/DihedralGroupD4.html)</sup>

## Notation and history

The group is written D₄ by authors who index dihedral groups by the number of vertices of the polygon, and D₈ by authors who index by the order; both conventions appear in the literature for the symmetries of the square.<sup>[12](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/06:_Groups_Acting_on_Sets/6.01:_Permutation_Groups)</sup> The name traces to [Felix Klein](https://www.edgepedia.org/felix-klein), who in the 1884 German original of his *Lectures on the Icosahedron* described the dihedral groups as rotation groups of dihedra and used the term Diedergruppe.<sup>[13](https://math.stackexchange.com/questions/1170883/history-of-dihedral-groups)</sup>

## Applications

The D₄ symmetry of the square appears throughout chemistry and physics. Square-planar molecules such as XeF₄ are analyzed with the related point groups D₄ and C₄v, whose character tables classify molecular vibrations and predict spectroscopic activity.<sup>[11](https://carstenjaeschke.com/essays/representation-theory.html)</sup><sup> • </sup><sup>[14](https://www.echem.energy.kyoto-u.ac.jp/gp/prob/prob3.pdf)</sup> In crystallography and materials modeling, structures built from squares exhibit D₄ invariance, and the group's representations constrain the allowed electronic band structures.<sup>[11](https://carstenjaeschke.com/essays/representation-theory.html)</sup><sup> • </sup><sup>[15](https://arxiv.org/abs/2310.12991)</sup> In particle physics, D₄ has been used as a discrete flavor symmetry in building SO(10) grand unified models.<sup>[16](https://arxiv.org/abs/1511.03166)</sup>

## References

1. [Dihedral Group D₄, Wolfram MathWorld](https://mathworld.wolfram.com/DihedralGroupD4.html)
2. [Groups of Small Order, A. Parris, UCSD](https://mathweb.ucsd.edu/~atparris/small_groups.html)
3. [GroupNames, Groups of Order 8, University of Bristol](https://people.maths.bris.ac.uk/~matyd/GroupNames/)
4. [Abstract Algebra, Chapter 4: Advanced Group Theory, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/algebra-ch04.html)
5. [Dihedral Group of Order 8, Colorado State University](https://www.math.colostate.edu/~jwilson/math/8/D8.html)
6. [Dihedral Groups, IAAWA, University of Tennessee](https://web.math.utk.edu/~dcartwr1/iaawa/section-dihedral-groups.html)
7. [Introduction to Groups, John D. Cobb, lecture notes](https://johndcobb.github.io/intro-abstract-algebra-notes/lectures/01-introduction-to-groups)
8. [Modern Algebra makeup exam, D. Bayer, Columbia University (2009)](https://www.math.columbia.edu/~bayer/S09/ModernAlgebra/Makeup1-ModernAlgebra-S09.pdf)
9. [Groups Up To Order Eight, P. Buser, Stanford notes](https://crypto.stanford.edu/pbc/notes/group/ordereight.html)
10. [Character Table, Wolfram MathWorld](https://mathworld.wolfram.com/CharacterTable.html)
11. [Representation Theory of the Square: D₄, Subgroups, and Decomposition, Carsten Jaeschke](https://carstenjaeschke.com/essays/representation-theory.html)
12. [Permutation Groups, Mathematics LibreTexts (Bogart)](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/06:_Groups_Acting_on_Sets/6.01:_Permutation_Groups)
13. [History of Dihedral Groups?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/1170883/history-of-dihedral-groups)
14. [Symmetry of Molecules and Group Theory, Kyoto University eChem](https://www.echem.energy.kyoto-u.ac.jp/gp/prob/prob3.pdf)
15. [DFT Studies of 2D Materials Inspired by Lie Algebras, arXiv:2310.12991](https://arxiv.org/abs/2310.12991)
16. [Building SO(10)-models with D₄ symmetry, arXiv:1511.03166](https://arxiv.org/abs/1511.03166)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Finite groups and classification › Subgroup structure and Sylow theory*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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