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

Up to 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).

Partition	Verbal description of cycle type	Representative element	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element orders
1 + 1 + 1 + 1 + 1 + 1	six fixed points	$()$ -- the identity element	1	$\frac{6!}{(1{)}^6(6!)}$	even; no	1
2 + 1 + 1 + 1 + 1	transposition, four fixed points	$(1,2)$	15	$\frac{6!}{(2)(1{)}^4(4!)}$	odd	2
3 + 1 + 1 + 1	one 3-cycle, three fixed points	$(1,2,3)$	40	$\frac{6!}{(3)(1{)}^3(3!)}$	even; no	3
4 + 1 + 1	one 4-cycle, two fixed points	$(1,2,3,4)$	90	$\frac{6!}{(4)(1{)}^2(2!)}$	odd	4
5 + 1	one 5-cycle, one fixed point	$(1,2,3,4,5)$	144	$\frac{6!}{(5)(1)}$	even; yes; yes	5
3 + 2 + 1	one 3-cycle, one 2-cycle, one fixed point	$(1,2,3)(4,5)$	120	$\frac{6!}{(3)(2)(1)}$	odd	6
2 + 2 + 1 + 1	double transposition: two 2-cycles, two fixed points	$(1,2)(3,4)$	45	$\frac{6!}{(2{)}^2(2!)(1{)}^2(2!)}$	even; no	2
2 + 2 + 2	triple transposition	$(1,2)(3,4)(5,6)$	15	$\frac{6!}{(2{)}^3(3!)}$	odd	2
4 + 2	one 4-cycle, one 2-cycle	$(1,2,3,4)(5,6)$	90	$\frac{6!}{(4)(2)}$	even; no	4
3 + 3	two 3-cycles	$(1,2,3)(4,5,6)$	40	$\frac{6!}{(3{)}^2(2!)}$	even; no	3
6	one 6-cycle	$(1,2,3,4,5,6)$	120	$\frac{6!}{6}$	odd	6

Up to automorphism

The outer automorphism group has order two, and it swaps some conjugacy classes. Below are the equivalence classes up to automorphisms.

Partitions for cycle types in one automorphism class	Representative elements for each	Size of each conjugacy class	Number of classes	Total size	Element orders
1 + 1 + 1 + 1 + 1 + 1	$()$	1	1	1	1
2 + 1 + 1 + 1 + 1, 2 + 2 + 2	$(1,2)$, $(1,2)(3,4)(5,6)$	15	2	30	2
3 + 1 + 1 + 1, 3 + 3	$(1,2,3)$, $(1,2,3)(4,5,6)$	40	2	80	3
4 + 1 + 1, 4 + 2	$(1,2,3,4)$, $(1,2,3,4)(5,6)$	90	2	180	4
5 + 1	$(1,2,3,4,5)$	144	1	144	5
3 + 2 + 1, 6	$(1,2,3)(4,5)$, $(1,2,3,4,5,6)$	120	2	240	6
2 + 2 + 1 + 1	$(1,2)(3,4)$	45	1	45	2

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.