Symmetric group:S6

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Definition

The symmetric group $S_6$, called the symmetric group of degree six, is defined in the following equivalent ways:

• It is the symmetric group on a set of size six. Other equivalent definitions include:
• It is the symplectic group $Sp(4,2)$, and hence also the projective symplectic group $PSp(4,2)$.

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.

GAP implementation

Group ID

This finite group has order 720 and has ID 763 among the groups of order 720 in GAP's SmallGroup library. For context, there are groups of order 720. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(720,763)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(720,763);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [720,763]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.