Linear representation theory of groups of order 8

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 8.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 8

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

To understand these in a broader context, see linear representation theory of groups of prime-cube order | linear representation theory of groups of order 2^n

Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

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 $p^k, 0 \le k \le 4$, 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.

Group	GAP ID second part	Hall-Senior number	Nilpotency class	Degrees as list	Number of irreps of degree 1 (= order of abelianization)	Number of irreps of degree 2	Total number of irreps (= 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

Splitting field

TERMINOLOGY AND FACTS TO CHECK AGAINST:
Terminology: field generated by character values (unique, cyclotomic) | minimal splitting field (not necessarily unique or cyclotomic) | splitting field | sufficiently large field (minimal choice is unique and cyclotomic)
Facts: sufficiently large implies splitting | splitting not implies sufficiently large | field generated by character values need not be a splitting field | field generated by character values is splitting field implies it is the unique minimal splitting field | minimal splitting field need not be unique | minimal splitting field need not be cyclotomic

Characteristic zero case

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

Field	Cyclotomic extension of rationals?	Real?	Degree of extension over $\mathbb{Q}$	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)
$\mathbb{Q}$	Yes	Yes	1	dihedral group:D8, elementary abelian group:E8	3, 5	dihedral group:D8, elementary abelian group:E8	3, 5
$\mathbb{Q}(i)$	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
$\mathbb{Q}(i,\sqrt{2})$	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

Field	Cyclotomic extension of rationals?	Real?	Degree of extension over $\mathbb{Q}$	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)
$\mathbb{Q}$	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
$\mathbb{Q}(i)$	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
$\mathbb{Q}(i,\sqrt{2})$	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

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 $t^d - 1$ splitting where $d$ is the 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 $2$.

Group	GAP ID	Polynomial that should split for it to be a splitting field	Condition for finite field with $q$ elements ($q$ odd)
cyclic group:Z8	1	$t^4 + 1$	$8$ divides $q - 1$
direct product of Z4 and Z2	2	$t^2 + 1$	$4$ divides $q - 1$
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 $-1$ 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

TERMINOLOGY AND FACTS TO CHECK AGAINST:
Terminology: ring generated by character values | minimal ring of realization of irreducible representations
Facts: linear representation is realizable over principal ideal domain iff it is realizable over field of fractions

Characteristic zero

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

General case

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

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Smallest set of values

Group	GAP ID	Set of character values	Minimal size set of values of matrix entries in suitable collection of representations
cyclic group:Z8	1	$\{ 1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4} \}$	$\{ 1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4} \}$
direct product of Z4 and Z2	2	$\{ 1, -1, i,-i \}$	$\{ 1, -1, i,-i \}$
dihedral group:D8	3	$\{ 1,-1,2,-2,0 \}$	$\{ 1,0,-1 \}$
quaternion group	4	$\{ 1,-1,2,-2,0 \}$	$\{ 1,0,-1,i,-i \}$ or $\{ 1,0,-1,\sqrt{2}i,-\sqrt{2}i \}$
elementary abelian group:E8	5	$\{ 1,-1\}$	$\{ 1,-1 \}$

Action of automorphisms and endomorphisms

Orbits under automorphism group

FACTS TO CHECK AGAINST for action of automorphism group on irreducible representations and degrees of irreducible representations
number of orbits of irreducible representations equals number of orbits under automorphism group | number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group
cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

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.

Group	Automorphism group	Outer automorphism group	Orbit sizes of degree 1 representations Sum of squares of degrees of irreps in each orbit	Orbit sizes of degree 2 representations Sum of squares of degrees of irreps in each orbit	Total number of irreps 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 1,1,2,4	no orbits	8 8	4
direct product of Z4 and Z2	dihedral group:D8	dihedral group:D8	1,1,2,4 1,1,2,4	no orbits	8 8	4
dihedral group:D8	dihedral group:D8	cyclic group:Z2	1,1,2 1,1,2	1 4	5 8	4
quaternion group	symmetric group:S4	symmetric group:S3	1,3 1,3	1 4	5 8	3
elementary abelian group:E8	projective special linear group:PSL(3,2)	projective special linear group:PSL(3,2)	1,7 1,7	no orbits	8 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.

Group	Field generated by character values	Galois group over rationals	Orbit sizes of degree 1 representations Sum of squares of degrees of irreps in each orbit	Orbit sizes of degree 2 representations Sum of squares of degrees of irreps in each orbit	Total number of irreps Sum of squares of degrees of irreps in each orbit	Total number of orbits under Galois group = 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	$\mathbb{Q}(i,\sqrt{2})$	Klein four-group	1,1,2,4 1,1,2,4	no orbits	8 8	4	2
direct product of Z4 and Z2	$\mathbb{Q}(i)$	cyclic group:Z2	1,1,1,1,2,2 1,1,1,1,2,2	no orbits	8 8	6	4
dihedral group:D8	$\mathbb{Q}$	trivial group	1,1,1,1 1,1,1,1	1 4	5 8	5	5
quaternion group	$\mathbb{Q}$	trivial group	1,1,1,1 1,1,1,1	1 4	5 8	5	5
elementary abelian group:E8	$\mathbb{Q}$	trivial group	1,1,1,1,1,1,1,1 1,1,1,1,1,1,1,1	no orbits	8 8	8	8

Relation with other orders

Divisors of the order

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

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