View source for Linear representation theory of groups of order 8

{{all given order groups-specific information|
order = 8|
information type = linear representation theory|
connective = of}}

{| class="sortable" border="1"
! Group !! GAP ID second part !! Hall-Senior number !! [[Nilpotency class]] !! Linear representation theory page
|-
| [[cyclic group:Z8]] || 1 || 3 || 1 || [[linear representation theory of cyclic group:Z8]], see also [[linear representation theory of cyclic groups]]
|-
| [[direct product of Z4 and Z2]] || 2 || 2 || 1 || [[linear representation theory of direct product of Z4 and Z2]]
|-
| [[dihedral group:D8]] || 3 || 4 || 2 || [[linear representation theory of dihedral group:D8]]
|-
| [[quaternion group]] || 4 || 5 || 2 || [[linear representation theory of quaternion group]]
|-
| [[elementary abelian group:E8]] || 5 || 1 || 1 || [[linear representation theory of elementary abelian group:E8]]
|}

{{quotation|To understand these in a broader context, see [[linear representation theory of groups of prime-cube order]] <nowiki>|</nowiki> [[linear representation theory of groups of order 2^n]]}}

==Degrees of irreducible representations==

{{irrep degrees facts to check against}}

The following sets of degrees of irreducible representations works over any [[splitting field]] not of characteristic two.

See also [[nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order]]. This says that for groups of order <math>p^k, 0 \le k \le 4</math>, the [[nilpotency class]] of the group, and the order, together determine the degrees of irreducible representations. In particular, for groups of order 8, there are only two cases: abelian, where all the irreducible representations have degree 1, and the class two groups, where there is one irreducible representation of degree 2.
<section begin="irrep-degrees-list"/>
{| class="sortable" border="1"
! Group !! GAP ID second part !! Hall-Senior number !! [[Nilpotency class]] !! Degrees as list !! Number of irreps of degree 1 ([[number of one-dimensional representations equals order of abelianization|= order of abelianization]]) !! Number of irreps of degree 2 !! Total number of irreps ([[number of irreducible representations equals number of conjugacy classes|= number of conjugacy classes]])
|-
| [[cyclic group:Z8]] || 1 || 3 || 1 || 1,1,1,1,1,1,1,1 || 8 || 0 || 8
|-
| [[direct product of Z4 and Z2]] || 2 || 2 || 1 || 1,1,1,1,1,1,1,1 || 8 || 0 || 8
|-
| [[dihedral group:D8]] || 3 || 4 || 2 || 1,1,1,1,2 || 4 || 1 || 5
|-
| [[quaternion group]] || 4 || 5 || 2 || 1,1,1,1,2 || 4 || 1 || 5
|-
| [[elementary abelian group:E8]] || 5 || 1 || 1 || 1,1,1,1,1,1,1,1 || 8 || 0 || 8
|}
<section end="irrep-degrees-list"/>

==Splitting field==

{{splitting field facts to check against}}

===Characteristic zero case===

{| class="sortable" border="1"
! Group !! GAP ID !! [[Field generated by character values]] !! Degree of extension over <math>\mathbb{Q}</math> !! Smallest field of realization of representations (i.e., [[minimal splitting field]]) in characteristic zero !! Degree of extension over <math>\mathbb{Q}</math> !! Minimal [[sufficiently large field]] !! Degree of extension over <math>\mathbb{Q}</math> !! Comment
|-
| [[cyclic group:Z8]] || 1 || <math>\mathbb{Q}(e^{\pi i/4}) = \mathbb{Q}(i,\sqrt{2})</math> || 4 || <math>\mathbb{Q}(e^{\pi i/4}) = \mathbb{Q}(i,\sqrt{2})</math> || 4 || <math>\mathbb{Q}(e^{\pi i/4}) = \mathbb{Q}(i,\sqrt{2})</math>  || 4 || abelian, so all fields coincide
|-
| [[direct product of Z4 and Z2]] || 2 || <math>\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)</math> || 2 || <math>\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)</math> || 2 || <math>\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)</math> || 2 || abelian, so all fields coincide
|-
| [[dihedral group:D8]] || 3 || <math>\mathbb{Q}</math> || 1 || <math>\mathbb{Q}</math> || 1 || <math>\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)</math> || 2 || [[splitting not implies sufficiently large]] -- the minimal splitting field is strictly smaller than the minimal sufficiently large field.
|-
| [[quaternion group]] || 4 || <math>\mathbb{Q}</math> || 1 || <math>\mathbb{Q}(i)</math> or <math>\mathbb{Q}(\sqrt{2}i)</math> or <math>\mathbb{Q}(\sqrt{-m^2 - 1})</math> where <math>m \in \mathbb{Q}</math> (among other possibilities). Follows from the fact that <math>\mathbb{Q}(\alpha,\beta)</math> is a splitting field if <math>\alpha^2 + \beta^2 = -1</math>. || 2 || <math>\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)</math> || 2 || [[minimal splitting field need not be unique]], [[splitting not implies sufficiently large]], [[minimal splitting field need not be cyclotomic]]
|-
| [[elementary abelian group:E8]] || 5 || <math>\mathbb{Q}</math> || 1 || <math>\mathbb{Q}</math> || 1 || <math>\mathbb{Q}</math> || 1 || abelian, so all fields coincide.
|}

===Grouping by minimal splitting field===

Note that since [[minimal splitting field need not be unique]], some groups have multiple minimal splitting fields. Among the groups of order 8, the only group with multiple minimal splitting fields is the [[quaternion group]].

{| class="sortable" border="1"
! Field !! Cyclotomic extension of rationals? !! Real? !! Degree of extension over <math>\mathbb{Q}</math> !! Groups for which this is a [[minimal splitting field]] !! GAP IDs (second part) !! Groups for which this is a [[splitting field]] (not necessarily minimal) !! GAP IDs (second part)
|-
| <math>\mathbb{Q}</math> || Yes || Yes || 1 || [[dihedral group:D8]], [[elementary abelian group:E8]] || 3, 5 || [[dihedral group:D8]], [[elementary abelian group:E8]] || 3, 5
|-
| <math>\mathbb{Q}(i)</math> || Yes || No || 2 || [[direct product of Z4 and Z2]], [[quaternion group]] || 2, 4 || [[direct product of Z4 and Z2]], [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 2, 3, 4, 5
|-
| <math>\mathbb{Q}(i,\sqrt{2})</math> || Yes || No || 4 || [[cyclic group:Z8]] || 1 || [[cyclic group:Z8]], [[direct product of Z4 and Z2]], [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 1,2,3,4,5
|}

===Grouping by field generated by character values===

{| class="sortable" border="1"
! Field !! Cyclotomic extension of rationals? !! Real? !! Degree of extension over <math>\mathbb{Q}</math> !! Groups for which this is the field generated by character values !! GAP IDs (second part) !! Groups for which this contains the field generated by character values !! GAP IDs (second part)
|-
| <math>\mathbb{Q}</math> || Yes || Yes || 1 || [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 3, 4, 5 || [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 3, 4, 5
|-
| <math>\mathbb{Q}(i)</math> || Yes || No || 2 || [[direct product of Z4 and Z2]] || 2 || [[direct product of Z4 and Z2]], [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 2, 3, 4, 5
|-
| <math>\mathbb{Q}(i,\sqrt{2})</math> || Yes || No || 4 || [[cyclic group:Z8]] || 1 || [[cyclic group:Z8]], [[direct product of Z4 and Z2]], [[dihedral group:D8]], [[quaternion group]], [[elementary abelian group:E8]] || 1,2,3,4,5
|}

===Rationals and reals: properties===

{| class="sortable" border="1"
! Group !! GAP ID !! Hall-Senior number !! [[nilpotency class]] !![[rational representation group]] (all representations realized over rationals)? !! [[rational group]] (all characters take rational values)? !! [[real-representation group]] (all representations realized over reals)? !! [[ambivalent group]] (all characters take real values)?
|-
| [[cyclic group:Z8]] || 1 || 3 || 1 || No || No || No || No
|-
| [[direct product of Z4 and Z2]] || 2 || 2 || 1 ||  No || No || No || No
|-
| [[dihedral group:D8]] || 3 || 4 || 2 || Yes || Yes || Yes || Yes
|-
| [[quaternion group]] || 4 || 5 || 2 || No || Yes || No || Yes
|-
| [[elementary abelian group:E8]] || 5 || 1 || 1 || Yes || Yes || Yes || Yes
|}

===General case===

Note that because [[sufficiently large implies splitting]], the polynomial <math>t^d - 1</math> splitting where <math>d</math> is the [[exponent of a group|exponent]] of the group is a sufficient condition for being a [[splitting field]]. However, it is not a necessary condition in general. For groups of order 8, the condition turns out to be necessary except in the case of [[dihedral group:D8]] and [[quaternion group]].

Here, we consider fields of characteristic not equal to <math>2</math>.

{| class="sortable" border="1"
! Group !! GAP ID !! Polynomial that should split for it to be a splitting field !! Condition for finite field with <math>q</math> elements (<math>q</math> odd)
|-
| [[cyclic group:Z8]] || 1 || <math>t^4 + 1</math> || <math>8</math> divides <math>q - 1</math>
|-
| [[direct product of Z4 and Z2]] || 2 || <math>t^2 + 1</math> || <math>4</math> divides <math>q - 1</math>
|-
| [[dihedral group:D8]]  || 3 || -- || none; any field works
|-
| [[quaternion group]] || 4 || cannot be expressed in terms of a single polynomial. Sufficient condition: any field (characteristic not 2) in which <math>-1</math> is a sum of two squares is a splitting field. Unclear if this is also a necessary condition. || none; any field works, because [[every element of a finite field is expressible as a sum of two squares]]
|-
| [[elementary abelian group:E8]] || 5 || -- || none; any field works
|}

==Ring of realization==

{{ring of realization facts to check against}}

===Characteristic zero===

{| class="sortable" border="1"
! Group !! GAP ID !! Ring generated by character values !! Degree of extension over <math>\mathbb{Z}</math> !! Smallest ring of realization of representations !! Degree of extension over <math>\mathbb{Z}</math>
|-
| [[cyclic group:Z8]] || 1 || <math>\mathbb{Z}[e^{\pi i/4}]</math> || 4 || <math>\mathbb{Z}[e^{\pi i/4}]</math> || 4
|-
| [[direct product of Z4 and Z2]] || 2 || <math>\mathbb{Z}[i]</math> || 2 || <math>\mathbb{Z}[i]</math> || 2
|-
| [[dihedral group:D8]] || 3 || <math>\mathbb{Z}</math> || 1 || <math>\mathbb{Z}</math> || 1
|-
| [[quaternion group]] || 4 || <math>\mathbb{Z}</math> || 1 || <math>\mathbb{Z}[i]</math> or <math>\mathbb{Z}[\sqrt{2}i]</math> or <math>\mathbb{Z}[\sqrt{-m^2 - 1}]</math> where <math>m \in \mathbb{Z}</math>. Follows from the fact that <math>\mathbb{Z}[\alpha,\beta]</math> is a ring of realization if <math>\alpha^2 + \beta^2 = -1</math> || 2
|-
| [[elementary abelian group:E8]] || 5 || <math>\mathbb{Z}</math> || 1 || <math>\mathbb{Z}</math> || 1
|}

===General case===

Here, we consider rings of characteristic either zero or a prime power <math>p^k</math>, where <math>p</math> is odd.

{{fillin}}

===Smallest set of values===

{| class="sortable" border="1"
! Group !! GAP ID !! Set of character values !! Minimal size set of values of matrix entries in suitable collection of representations
|-
| [[cyclic group:Z8]] || 1 || <math>\{ 1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4} \}</math> || <math>\{ 1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4} \}</math>
|-
| [[direct product of Z4 and Z2]] ||2 ||<math>\{ 1, -1, i,-i \}</math> || <math>\{ 1, -1, i,-i \}</math>
|-
| [[dihedral group:D8]] || 3 || <math>\{ 1,-1,2,-2,0 \}</math> || <math>\{ 1,0,-1 \}</math>
|-
| [[quaternion group]] || 4 || <math>\{ 1,-1,2,-2,0 \}</math> || <math>\{ 1,0,-1,i,-i \}</math> or <math>\{ 1,0,-1,\sqrt{2}i,-\sqrt{2}i \}</math>
|-
| [[elementary abelian group:E8]] || 5 || <math>\{ 1,-1\}</math> || <math>\{ 1,-1 \}</math>
|}

==Action of automorphisms and endomorphisms==

===Orbits under automorphism group===

{{irrep degrees automorphism group action facts to check against}}

We provide here the structure and sizes of orbits under the action of the automorphism group. The information presented below is valid in characteristic zero.

{| class="sortable" border="1"
! Group !! Automorphism group !! Outer automorphism group !! Orbit sizes of degree 1 representations<br>Sum of squares of degrees of irreps in each orbit !! Orbit sizes of degree 2 representations<br>Sum of squares of degrees of irreps in each orbit !! Total number of irreps<br>Sum of squares of degrees of irreps in each orbit !! Total number of orbits under automorphism group
|-
| [[cyclic group:Z8]] || [[Klein four-group]] || [[Klein four-group]] || 1,1,2,4<br>1,1,2,4 || no orbits || 8<br>8 || 4
|-
| [[direct product of Z4 and Z2]] || [[dihedral group:D8]] || [[dihedral group:D8]] || 1,1,2,4<br>1,1,2,4 || no orbits || 8<br>8 || 4
|-
| [[dihedral group:D8]] || [[dihedral group:D8]] || [[cyclic group:Z2]] || 1,1,2<br>1,1,2 || 1<br>4  || 5<br>8 || 4
|-
| [[quaternion group]] || [[symmetric group:S4]] || [[symmetric group:S3]] || 1,3<br>1,3 || 1<br>4 || 5<br>8 || 3
|-
| [[elementary abelian group:E8]] || [[projective special linear group:PSL(3,2)]] || [[projective special linear group:PSL(3,2)]] || 1,7<br>1,7 || no orbits || 8<br>8 || 2
|}

==Action of Galois automorphisms==

We provide here the structure and sizes of orbits under the action of Galois automorphisms. The information presented below is valid in characteristic zero.

{| class="sortable" border="1"
! Group !! Field generated by character values !! Galois group over rationals !! Orbit sizes of degree 1 representations<br>Sum of squares of degrees of irreps in each orbit !! Orbit sizes of degree 2 representations<br>Sum of squares of degrees of irreps in each orbit !! Total number of irreps<br>Sum of squares of degrees of irreps in each orbit !! Total number of orbits under Galois group = [[Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy|number of irreducible representations over rationals = number of equivalence classes under rational conjugacy]] !! Number of fixed points under Galois group = number of irreducible representations over complex numbers with rational character values
|-
| [[cyclic group:Z8]] || <math>\mathbb{Q}(i,\sqrt{2})</math> || [[Klein four-group]] || 1,1,2,4<br>1,1,2,4 || no orbits || 8<br>8 || 4 || 2
|-
| [[direct product of Z4 and Z2]] || <math>\mathbb{Q}(i)</math> || [[cyclic group:Z2]] || 1,1,1,1,2,2<br>1,1,1,1,2,2 || no orbits || 8<br>8 || 6 || 4
|-
| [[dihedral group:D8]] || <math>\mathbb{Q}</math> || [[trivial group]] || 1,1,1,1<br>1,1,1,1 || 1<br>4  || 5<br>8 || 5 || 5
|-
| [[quaternion group]] || <math>\mathbb{Q}</math> || [[trivial group]] ||  1,1,1,1<br>1,1,1,1 || 1<br>4 || 5<br>8 || 5 || 5
|-
| [[elementary abelian group:E8]] || <math>\mathbb{Q}</math> || [[trivial group]] || 1,1,1,1,1,1,1,1<br>1,1,1,1,1,1,1,1 || no orbits || 8<br>8 || 8 || 8
|}

==Relation with other orders==

===Divisors of the order===

{| class="sortable" border="1"
! Divisor !! Quotient value !! Information on linear representation theory !! Relationship (subgroup perspective) !! Relationship (quotient perspective)
|-
| 2 || 4 || [[linear representation theory of cyclic group:Z2]] || ||
|-
| 4 || 2 || [[linear representation theory of groups of order 4]] ([[linear representation theory of cyclic group:Z4]], [[linear representation theory of Klein four-group]]) || ||
|}

===Multiples of the order===

{| class="sortable" border="1"
! Multiplier (other factor) !! Multiple !! Information on linear representation theory !! Relationship (subgroup perspective) !! Relationship (quotient perspective)
|-
| 2 || 16 || [[linear representation theory of groups of order 16]] || ||
|-
| 3 || 24 || [[linear representation theory of groups of order 24]] || ||
|-
| 4 || 32 || [[linear representation theory of groups of order 32]] || ||
|-
| 5 || 40 || [[linear representation theory of groups of order 40]] || ||
|-
| 6 || 48 || [[linear representation theory of groups of order 48]] || ||
|-
| 7 || 56 || [[linear representation theory of groups of order 56]] || ||
|-
| 8 || 64 || [[linear representation theory of groups of order 64]] || ||
|-
| 9 || 72 || [[linear representation theory of groups of order 72]] || ||
|-
| 10 || 80 || [[linear representation theory of groups of order 80]] || ||
|-
| 11 || 88 || [[linear representation theory of groups of order 88]] || ||
|-
| 12 || 96 || [[linear representation theory of groups of order 96]] || ||
|}