Symmetric group:S6

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This particular group is a finite group of order: 720

Definition

The symmetric group $S_6$, called the symmetric group of degree six, is defined as the symmetric group on a set of size six. Other equivalent definitions include:

• It is the projective general linear group $PGL(2,9)$.

Elements

Upto conjugacy

For convenience, we take the underlying set here as $\{1,2,3,4,5,6\}$.

There are eleven conjugacy classes, corresponding to the unordered integer partitions of $6$ (for more information, refer cycle type determines conjugacy class):

1. $6=1+1+1+1+1+1$, i.e., six cycles of size one: The identity element. (1)
2. $6=2+1+1+1+1$, i.e., one $2$-cycle and four fixed points: The transpositions, such as $(1,2)$. (15)
3. $6=3+1+1+1$, i.e., one $3$-cycle and three fixed points: The $3$-cycles, such as $(1,2,3)$. (40)
4. $6=4+1+1$: The $4$-cycles, such as $(1,2,3,4)$. (90)
5. $6=5+1$: The $5$-cycles, such as $(1,2,3,4,5)$. (144)
6. $6=6$: The $6$-cycles, such as $(1,2,3,4,5,6)$. (120)
7. $6=2+2+1+1$, i.e., two $2$-cycles, two fixed points: The double transpositions, such as $(1,2)(3,4)$. (45)
8. $6=2+2+2$, i.e., three $2$-cycles: The triple transpositions, such as $(1,2)(3,4)(5,6)$. (15)
9. $6=3+2+1$, i.e., one $3$-cycle, one $2$-cycle: Permutations such as $(1,2,3)(4,5)$.(120)
10. $6=3+3$, i.e., two $3$-cycles: Permutations such as $(1,2,3)(4,5,6)$. (40)
11. $6=4+2$: Permutations such as $(1,2,3,4)(5,6)$. (90)

Of these, types (1), (3), (5), (7), (10), (11) are conjugacy classes of even permutations -- together these form the alternating group of degree six. The remaining types: (2), (4), (6), (8), (9), are odd permutations.

Upto automorphism

Under automorphisms, the following types get merged:

• Types (2) and (8): The transposition $(1,2)$ is related by an outer automorphism to the triple transposition $(1,2)(3,4)(5,6)$.
• Types (3) and (10): The $3$-cycle $(1,2,3)$ is related by an outer automorphism to the permutation $(1,2,3)(4,5,6)$.
• Types (4) and (11): The $4$-cycle $(1,2,3,4)$ is related by an outer automorphism to the pemrutation $(1,2,3,4)(5,6)$.
• Types (6) and (9): The $6$-cycle $(1,2,3,4,5,6)$ is related by an outer automorphism to the permutation $(1,2,3)(4,5)$.

The types (1), (5), and (7) remain unaffected: these conjugacy classes are also automorphism classes.