Alternating Groups
The alternating group A_{n} is of order (½)n!. In these tables, elements are denoted as products of disjoint cycles. For example,
 C_{2}C_{2} = (1 2)(3 4)
 C_{3} = (1 2 3)
On this page:
 A_{4} ≅ T, the group of rotational symmetries of the tetrahedron
 A_{5} ≅ I, the group of rotational symmetries of the icosahedron
 A_{6}
A_{4}
A_{4}
 e
 C_{2}C_{2}
 C_{3}
 C_{3}^{2}
 notes

#
 1
 3
 4
 4
 A_{4}=12

A_{0}
 1
 1
 1
 1
 trivial rep

E
 2
 2
 1
 1
 not absolutely irreducible

T
 3
 1
 0
 0
 natural rep of tetrahedral group

 The natural permutation representation on 4 objects is \(A_0 + T\)
 The permutation representation on the 6 edges of the tetrahedron is \(A_0 + E + T\)
 The permutation representation on the 3 'diagonals' (joining midpoints of opposite edges) is \(A_0+E\). Each of these diagonals is fixed by the Klein 4group \(V_4\).
 The "oriented permutation" representation on the 6 edges of the tetrahedron is \(2T\)
 \(E\) is of complex type, and its complexification splits over \(\mathbf{C}\) as (1, 1, ω, ω²) ⊕ (1, 1, ω², ω), where \(\omega=\frac12(1+i\sqrt{3})\) is a cube root of unity.
A_{5}
A_{5}
 e
 C_{2}C_{2}
 C_{3}
 C_{5}
 C_{5}^{2}
 notes

#
 1
 15
 20
 12
 12
 A_{5}=60

A_{0}
 1
 1
 1
 1
 1
 trivial rep

T_{1}
 3
 1
 0
 \(\varphi^+\)
 \(\varphi^\)
 Symmetries of icosahedron

T_{2}
 3
 1_{}
 0
 \(\varphi^\)
 \(\varphi^+\)


G
 4
 0
 1
 1
 1


H
 5
 1
 1
 0
 0


 \(\varphi^+ = 2\cos(\pi/5) = \frac12(1+\sqrt5)\) (=golden ratio) and
\(\varphi^ = 2\cos(2\pi/5) = \frac12(1\sqrt5)\) ( = \((\varphi^+)^{1}\))
 All the representations are absolutely irreducible
 \(T_1\) and \(T_2\) are related by an outer automorphism of the group.
 The representation \(T := T_1 + T_2\) is irreducible over \(\mathbb{Q}\), but not (of course) absolutely irreducible; it becomes reducible over \(\mathbb{Q}[\sqrt5]\). It is also the restriction to \(A_5\) of the 6dimensional irreducible representation of \(S_5\), see here.
 \(A_5\) is the group of rotational symmetries of the regular icosahedron (and dodecahedron), so denoted I in Schoenflies. If C_{5} = (1 2 3 4 5) acts by rotations by \(2\pi/5\) then this geometric representation is T_{1}.
 The natural permutation representation on 5 objects is \(A_0 + G\)
 The permutation representation on the set of 12 vertices of the icosahedron is \(A_0 + T + H\)
 The permutation representation on the set of 20 vertices of the dodecahedron is \(A_0 + T + 2G + H\)
 The permutation representation on the set of 30 edges of either is \(A_0 + T + 2G + 3H\)
 The "oriented permutation" representation on the set of 30 edges of either is \(2(T + G + H)\)
 The permutation representation on the set of 6 diagonals of the icosahedron is \(A_0 + H\)
 The dodecahedron famously contains 5 inscribed tetrahedra (each formed by joining 4 nonadjacent vertices) (also 5 cubes). The resulting permutation representation is \(A_0 + G\).
A_{6}
A_{6}
 e
 C_{2}C_{2}
 C_{2}C_{4}
 C_{3}
 C_{3}C_{3}
 C_{5}
 C_{5}^{2}
 notes

#
 1
 45
 90
 40
 40
 72
 72
 A_{6}=360

A_{0}
 1
 1
 1
 1
 1
 1
 1
 trivial rep

H_{1}
 5
 1
 1
 2
 1
 0
 0


H_{2}
 5
 1
 1
 1
 2
 0
 0


L_{1}
 8
 0
 0
 1
 1
 \(\varphi^+\)
 \(\varphi^\)


L_{2}
 8
 0
 0
 1
 1
 \(\varphi^\)
 \(\varphi^+\)


M
 9
 1
 1
 0
 0
 1
 1


N
 10
 2
 0
 1
 1
 0
 0


 Names of reps are not standard
 \(\varphi^+\) and \(\varphi^\) are as for A_{5} above
 L_{1} + L_{2} is the restriction to A_{6} of the representation U of S_{6}.