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| {{group-specific information|
| | #redirect [[Linear representation theory of modular maximal-cyclic group:M16]] |
| information type = linear representation theory|
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| group = M16|
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| connective = of}}
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| This article discusses the linear representation theory of the group [[M16]], a group of order 16 given by the [[presentation]]:
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| <math>G = M_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle</math>
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| ==Summary==
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| {| class="sortable" border="1"
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| ! Item !! Value
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| | [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times)<br>[[maximum degree of irreducible representation|maximum]]: 2, [[lcm of degrees of irreducible representations|lcm]]: 2, [[number of irreducible representations equals number of conjugacy classes|number]]: 10, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 16
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| | [[Schur index]] values of irreducible representations || 1 (all of them)
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| |-
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| | smallest ring of realization (characteristic zero) || <math>\mathbb{Z}[i] = \mathbb{Z}[\sqrt{-1}] = \mathbb{Z}[t]/(t^2 + 1)</math> -- [[ring of Gaussian integers]]
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| | [[ring generated by character values]] (characteristic zero) || <math>\mathbb{Z}[2i] = \mathbb{Z}[\sqrt{-4}] = \mathbb{Z}[t]/(t^2 + 4)</math>
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| |-
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| | [[minimal splitting field]], i.e., smallest field of realization (characteristic zero) || <math>\mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)</math><br>Same as [[field generated by character values]], because all Schur index values are 1.
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| | condition for a field to be a [[splitting field]] || The characteristic should not be equal to 2, and the polynomial <math>t^2 + 1</math> should split.<br>For a finite field of size <math>q</math>, this is equivalent to saying that <math>q \equiv 1 \pmod 4</math>
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| | [[minimal splitting field]] in characteristic <math>p \ne 0,2</math> || Case <math>p \equiv 1 \pmod 4</math>: prime field <math>\mathbb{F}_p</math><br>Case <math>p \equiv 3 \pmod 4</math>: Field <math>\mathbb{F}_{p^2}</math>, quadratic extension of prime field
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| | smallest size splitting field || [[Field:F5]], i.e., the field with five elements.
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| | degrees of irreducible representations over the rational numbers || 1,1,1,1,2,2,4 (1 occurs 4 times, 2 occurs 2 times, 4 occurs 1 time)<br>[[Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy|number]]: 7
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| | orbits of irreducible representations over a [[splitting field]] under action of [[automorphism group]] || 2 orbits of size 1 of degree 1 representations, 3 orbits of size 2 of degree 1 representations, 1 orbit of size 2 of degree 2 representations.<br>[[number of orbits of irreducible representations equals number of orbits under automorphism group|number]]: 6
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| |}
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| ==Representations==
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| ===Summary information===
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| 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.
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| {| class="sortable" border="1"
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| ! Name of representation type !! Number of representations of this type !! [[Degree of a linear representation|Degree]] !! [[Schur index]] !! Criterion for field !! Kernel (a [[normal subgroup]] of [[M16]] comprising the elements that map to identity matrices; see [[subgroup structure of M16]])!! Quotient by kernel (on which it descends to a faithful representation) !! Characteristic 2?
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| | trivial || 1 || 1 || 1 || any || whole group || [[trivial group]] || works
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| |-
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| | sign, kernel a non-cyclic maximal subgroup || 1 || 1 || 1 || any || [[direct product of Z4 and Z2 in M16]] -- <math>\langle a^2, x \rangle</math> || [[cyclic group:Z2]] || works, same as trivial
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| | sign, kernel a cyclic maximal subgroup || 2 || 1 || 1 || any || [[Z8 in M16]] -- either <math>\langle a \rangle</math> or <math>\langle ax \rangle</math> || [[cyclic group:Z2]] || works, same as trivial
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| | representation with kernel <math>\langle a^2x \rangle</math> || 2 || 1 || 1 || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || [[non-central Z4 in M16]] || [[cyclic group:Z4]] || works, same as trivial
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| | representation with kernel <math>\langle a^4, x \rangle</math> || 2 || 1 || 1 || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || [[V4 in M16]] || [[cyclic group:Z4]] ||works, same as trivial
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| | [[faithful irreducible representation of M16]] || 2 || 2 || 1 || must contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> must split || trivial subgroup || [[M16]] || indecomposable but not irreducible
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| |}
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| Below are representations that are irreducible over a non-splitting field, but split over a splitting field.
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| {| class="sortable" border="1"
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| ! Name of representation type !! Number of representations of this type !! [[Degree of a linear representation|Degree]] !! Criterion for field !! What happens over a splitting field? !! Kernel !! Quotient by kernel (on which it descends to a faithful representation)
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| | two-dimensional representation with kernel <math>\langle a^2x \rangle</math> || 1 || 2 || must ''not'' contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> does not split || splits into the one-dimensional representations with kernel <math>\langle a^2x \rangle</math> || [[non-central Z4 in M16]] || [[cyclic group:Z4]]
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| | two-dimensional representation with kernel <math>\langle a^4,x \rangle</math> || 1 || 2 || must ''not'' contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> does not split || splits into the one-dimensional representations with kernel <math>\langle a^4,x \rangle</math> || [[Klein four-group]] || [[cyclic group:Z4]]
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| | four-dimensional representation || 1 || 4 || must ''not'' contain a primitive fourth root of unity, or equivalently, <math>t^2 + 1</math> does not split || splits into the two faithful irreducible representations of degree two || trivial subgroup || [[M16]]
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| |}
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| ==Character table==
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| {{character table facts to check against}}
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| Below is the character table over a [[splitting field]]. Here <math>i</math> denotes a square root of <matH>-1</math> in the field.
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| {| class="sortable" border="1"
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| ! Representation/conjugacy class and size !! <math>\{ e \}</math> (size 1) !! <math>\{ a^4 \}</math> (size 1) !! <math>\{ a^2 \}</math> (size 1) !! <math>\{ a^6 \}</math> (size 1) !! <math>\{ a, a^5 \}</math> (size 2) !! <math>\{ a^3, a^7 \}</math> (size 2) !! <math>\{ ax, a^5x \}</math> (size 2) !! <math>\{ a^3x, a^7x \}</math> (size 2) !! <math>\{ x, a^4x\}</math> (size 2) !! <math>\{ a^2x, a^6x \}</math> (size 2)
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| | trivial || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
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| | <math>\langle a^2,x \rangle</math>-kernel || 1 || 1 || 1 || 1 || -1 || -1 || -1 || -1 || 1 || 1
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| | <math>\langle a \rangle</math>-kernel || 1 || 1 || 1 || 1 || 1 || 1 || -1 || -1 || -1 || -1
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| | <math>\langle ax \rangle</math>-kernel || 1 || 1 || 1 || 1 || -1 || -1 || 1 || 1 || -1 || -1
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| | <math>\langle a^2x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>i</math> || <math>-i</math> || <math>-i</math> || <math>i</math> || -1 || 1
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| | <math>\langle a^2x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-i</math> || <math>i</math> || <math>i</math> || <matH>-i</math> || -1 || 1
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| | <math>\langle a^4,x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>i</math> || <math>-i</math> || <math>i</math> || <math>-i</math> || 1 || -1
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| | <math>\langle a^4,x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-i</math> || <math>i</math> || <math>-i</math> || <math>i</math> || 1 || -1
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| | [[faithful irreducible representation of M16]] (first)|| 2 || -2 || <math>2i</math> || <math>-2i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| | [[faithful irreducible representation of M16]] (second) || 2 || -2 || <math>-2i</math> || <math>2i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| |}
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| Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and dividing by the degree of the irreducible representation:
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| {| class="sortable" border="1"
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| ! Representation/conjugacy class and size !! <math>\{ e \}</math> (size 1) !! <math>\{ a^4 \}</math> (size 1) !! <math>\{ a^2 \}</math> (size 1) !! <math>\{ a^6 \}</math> (size 1) !! <math>\{ a, a^5 \}</math> (size 2) !! <math>\{ a^3, a^7 \}</math> (size 2) !! <math>\{ ax, a^5x \}</math> (size 2) !! <math>\{ a^3x, a^7x \}</math> (size 2) !! <math>\{ x, a^4x\}</math> (size 2) !! <math>\{ a^2x, a^6x \}</math> (size 2)
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| | trivial || 1 || 1 || 1 || 1 || 2 || 2 || 2 || 2 || 2 || 2
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| | <math>\langle a \rangle</math>-kernel || 1 || 1 || 1 || 1 || 2 || 2 || -2 || -2 || -2 || -2
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| | <math>\langle ax \rangle</math>-kernel || 1 || 1 || 1 || 1 || -2 || -2 || 2 || 2 || -2 || -2
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| | <math>\langle a^2,x \rangle</math>-kernel || 1 || 1 || 1 || 1 || -2 || -2 || -2 || -2 || 2 || 2
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| | <math>\langle a^2x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>2i</math> || <math>-2i</math> || <math>-2i</math> || <math>2i</math> || -2 || 2
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| | <math>\langle a^2x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-2i</math> || <math>2i</math> || <math>2i</math> || <math>-2i</math> || -2 || 2
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| | <math>\langle a^4,x \rangle</math>-kernel (first) || 1 || 1 || -1 || -1 || <math>2i</math> || <math>-2i</math> || <math>2i</math> || <math>-2i</math> || 2 || -2
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| | <math>\langle a^4,x \rangle</math>-kernel (second) || 1 || 1 || -1 || -1 || <math>-2i</math> || <math>2i</math> || <math>-2i</math> || <math>2i</math> || 2 || -2
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| | [[faithful irreducible representation of M16]] || 1 || -1 || <math>i</math> || <math>-i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| | [[faithful irreducible representation of M16]] (second) || 1 || -1 || <math>-i</math> || <math>i</math> || 0 || 0 || 0 || 0 || 0 || 0
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| |}
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| ==GAP implementation==
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| ===Degrees of irreducible representations===
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| The degrees of irreducible representation can be computed using the [[GAP:CharacterDegrees|CharacterDegrees]] function:
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| <pre>gap> CharacterDegrees(SmallGroup(16,6));
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| [ [ 1, 8 ], [ 2, 2 ] ]</pre>
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| ===Character table===
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| The character table can be computed using the [[GAP:Irr|Irr]] and [[GAP:CharacterTable|CharacterTable]] functions:
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| <pre>gap> Irr(CharacterTable(SmallGroup(16,6)));
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| [ Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, -1, 1, 1, 1, -1, -1, 1, 1, -1 ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, 1, -1, 1, 1, -1, 1, -1, 1, -1 ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, -1, -1, 1, 1, 1, -1, -1, 1, 1 ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, E(4), 1, -1, 1, E(4), -E(4), -1, -1, -E(4) ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, -E(4), 1, -1, 1, -E(4), E(4), -1, -1, E(4) ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, E(4), -1, -1, 1, -E(4), -E(4), 1, -1, E(4) ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 1, -E(4), -1, -1, 1, E(4), E(4), 1, -1, -E(4) ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 2, 0, 0, 2*E(4), -2, 0, 0, 0, -2*E(4), 0 ] ),
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| Character( CharacterTable( <pc group of size 16 with 4 generators> ),
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| [ 2, 0, 0, -2*E(4), -2, 0, 0, 0, 2*E(4), 0 ] ) ]</pre>
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| A nicer display can be achieved using the Display function:
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| <pre>gap> Display(CharacterTable(SmallGroup(16,6)));
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| CT3
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| 2 4 3 3 4 4 3 3 3 4 3
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| 1a 8a 2a 4a 2b 8b 8c 4b 4c 8d
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| X.1 1 1 1 1 1 1 1 1 1 1
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| X.2 1 -1 1 1 1 -1 -1 1 1 -1
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| X.3 1 1 -1 1 1 -1 1 -1 1 -1
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| X.4 1 -1 -1 1 1 1 -1 -1 1 1
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| X.5 1 A 1 -1 1 A -A -1 -1 -A
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| X.6 1 -A 1 -1 1 -A A -1 -1 A
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| X.7 1 A -1 -1 1 -A -A 1 -1 A
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| X.8 1 -A -1 -1 1 A A 1 -1 -A
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| X.9 2 . . B -2 . . . -B .
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| X.10 2 . . -B -2 . . . B .
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| A = E(4)
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| = ER(-1) = i
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| B = 2*E(4)
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| = 2*ER(-1) = 2i</pre>
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| ===Irreducible representations===
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| The irreducible representations can be computed explicitly using the [[GAP:IrreducibleRepresentations|IrreducibleRepresentations]] function:
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| <pre>gap> IrreducibleRepresentations(SmallGroup(16,6));
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| [ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ]
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| ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ 1 ] ], [ [ -1 ] ],
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| [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ -1 ] ], [ [ -1 ] ],
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| [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ -1 ] ], [ [ -1 ] ],
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| [ [ 1 ] ] ],
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, E(4) ], [ 1, 0 ] ], [ [ 1, 0 ],
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| [ 0, -1 ] ], [ [ E(4), 0 ], [ 0, E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ]
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| ,
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| Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, -E(4) ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0,
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| -1 ] ], [ [ -E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ]
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| ] ]</pre>
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