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	Linear representation theory page
cyclic group:Z8	1	3	linear representation theory of cyclic group:Z8, see also linear representation theory of cyclic groups
direct product of Z4 and Z2	2	2	linear representation theory of direct product of Z4 and Z2
dihedral group:D8	3	4	linear representation theory of dihedral group:D8
quaternion group	4	5	linear representation theory of quaternion group
elementary abelian group:E8	5	1	linear representation theory of elementary abelian group:E8

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 field not of characteristic two. See also

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	Degrees as list	Number of irreps of degree 1	Number of irreps of degree 2
cyclic group:Z8	1	1,1,1,1,1,1,1,1	8	0
direct product of Z4 and Z2	2	1,1,1,1,1,1,1,1	8	0
dihedral group:D8	3	1,1,1,1,2	4	1
quaternion group	4	1,1,1,1,2	4	1
elementary abelian group:E8	5	1,1,1,1,1,1,1,1	8	0

Field of realization (characteristic zero)

Smallest field of realization

Group	GAP ID	Field generated by character values	Degree of extension over $\mathbb{Q}$	Smallest field of realization of representations	Degree of extension over $\mathbb{Q}$
cyclic group:Z8	1	$\mathbb{Q}(e^{\pi i/4})$	4	$\mathbb{Q}(e^{\pi i/4})$	4
direct product of Z4 and Z2	2	$\mathbb{Q}(i)$	2	$\mathbb{Q}(i)$	2
dihedral group:D8	3	$\mathbb{Q}$	1	$\mathbb{Q}$	1
quaternion group	4	$\mathbb{Q}$	1	$\mathbb{Q}(i)$	2
elementary abelian group:E8	5	$\mathbb{Q}$	1	$\mathbb{Q}$	1

Smallest ring of realization

Group	GAP ID	Ring generated by character values	Degree of extension over $\mathbb{Z}$	Smallest field 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	$\mathbb{Z}[i]$	2
elementary abelian group:E8	5	$\mathbb{Z}$	1	$\mathbb{Z}$	1

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	$\{1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4}\}$	$\{1,-1,i,-i,e^{\pi i/4},e^{3\pi i/4},e^{-\pi i/4},e^{-3\pi i/4}\}$
direct product of Z4 and Z2	$\{1,-1,i,-i\}$	$\{1,-1,i,-i\}$
dihedral group:D8	3	$\{1,-1,2,-2,0\}$	$\{1,0,-1\}$
quaternion group	4	$\{1,-1,2,-2,0\}$	$\{1,0,-1,i,-i\}$
elementary abelian group:E8	5	$\{1,-1\}$	$\{1,-1\}$

Rationals and reals

Group	GAP ID	rational representation group (all representations realized over rationals)?	rational group (all characters take rational values)?	ambivalent group (all characters take real values)?
cyclic group:Z8	1	No	No	No
direct product of Z4 and Z2	2	No	No	No
dihedral group:D8	3	Yes	Yes	Yes
quaternion group	4	No	Yes	Yes
elementary abelian group:E8	5	Yes	Yes	Yes