Linear representation theory of semidihedral group:SD16

Summary
We shall use the semidihedral group of order 16 with the following presentation:

$$\langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^3 \rangle$$.

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Below are representations that are irreducible over a non-splitting field, but split over a splitting field.

Trivial representation
The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the $$1 \times 1$$ matrix $$(1)$$:

Sign representation with kernel $$\langle a \rangle$$
This representation is a one-dimensional representation sending everything in the cyclic subgroup $$\langle a \rangle$$ (see Z8 in SD16) to $$(1)$$ and everything outside it to $$(-1)$$.

To keep the description short, we club together the cosets rather than having one row per element:

Sign representation with kernel $$\langle a^2, x \rangle$$
There is a sign representation with kernel $$\langle a^2, x\rangle$$ which is dihedral group:D8 (see D8 in SD16). Everything inside the subgroup goes to $$(1)$$ and everything outside the subgroup goes to $$(-1)$$.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Sign representation with kernel $$\langle a^2, ax \rangle$$
There is a sign representation with kernel $$\langle a^2, ax\rangle$$ which is quaternion group (see Q8 in SD16). Everything inside the subgroup goes to $$(1)$$ and everything outside the subgroup goes to $$(-1)$$.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Two-dimensional irreducible unfaithful representation
This representation has kernel equal to $$\langle a^4 \rangle$$ -- center of semidihedral group:SD16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Character table
 Below is the character table over a splitting field, where $$\sqrt{-2}$$ stands for any chosen square root of $$-2$$:

Degrees of irreducible representations
These can be computed using the CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(16,8)); [ [ 1, 4 ], [ 2, 3 ] ]

Character table
The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SmallGroup(16,8))); [ Character( CharacterTable(  ),   [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ), [ 1, -1, 1, 1, 1, -1, -1 ] ), Character( CharacterTable(  ),   [ 1, 1, -1, 1, 1, -1, -1 ] ), Character( CharacterTable(  ),   [ 1, -1, -1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ), [ 2, 0, 0, -2, 2, 0, 0 ] ), Character( CharacterTable(  ),   [ 2, 0, 0, 0, -2, -E(8)-E(8)^3, E(8)+E(8)^3 ] ), Character( CharacterTable(  ),   [ 2, 0, 0, 0, -2, E(8)+E(8)^3, -E(8)-E(8)^3 ] ) ]

A nicer tabular display can be achieved using the Display function:

gap> Display(CharacterTable(SmallGroup(16,8))); CT1

2 4  2  2  3  4  3  3

1a 4a 2a 4b 2b 8a 8b

X.1    1  1  1  1  1  1  1 X.2    1 -1  1  1  1 -1 -1 X.3    1  1 -1  1  1 -1 -1 X.4    1 -1 -1  1  1  1  1 X.5    2. . -2 2  .  . X.6     2. . . -2  A -A X.7    2. . . -2 -A  A

A = -E(8)-E(8)^3 = -ER(-2) = -i2

Irreducible representations
The irreducible representations can be computed using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SmallGroup(16,8)); [ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ] , Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, -E(8) ], [ -E(8)^3, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, E(8) ], [ E(8)^3, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] ]