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 $PSp(4,2)$.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	720	groups with same order	As $S_k,k=6:$ $k!=6!=6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=720$
exponent	60	groups with same order and exponent | groups with same exponent	As $S_k,k=6:$ $\mathrm{l}\mathrm{c}\mathrm{m}\{1,2,3,4,5,6\}=60$
derived length	--		not a solvable group
nilpotency class	--		not a nilpotent group
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Frattini-free group; see also symmetric groups are Frattini-free
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	$(1,2),(1,2,3,4,5,6)$; see also symmetric group on a finite set is 2-generated
subgroup rank of a group	3	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group	The group elementary abelian group:E8 can be embedded in this group as $⟨(1,2),(3,4),(5,6)⟩$
max-length of a group	6	groups with same order and max-length of a group | groups with same max-length of a group	This is a rare example of a small group whose max-length is less than the sum of the exponents of all prime divisors
composition length	2	groups with same order and composition length | groups with same composition length	The subgroup alternating group:A6 is simple and normal (see alternating groups are simple) and the quotient is simple (cyclic of order two)
chief length	2	groups with same order and chief length | groups with same chief length	The subgroup alternating group:A6 is simple and normal (see alternating groups are simple) and the quotient is simple (cyclic of order two)
number of subgroups	1455	groups with same order and number of subgroups | groups with same number of subgroups
number of conjugacy classes	11	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $S_k,k=6:$ the number of conjugacy classes is $p(k)=p(6)=11$, where $p$ is the number of unordered integer partitions; see cycle type determines conjugacy class.
number of conjugacy classes of subgroups	56	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

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	Partition in grouped form	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	Formula calculating element order
1 + 1 + 1 + 1 + 1 + 1	1 (6 times)	six fixed points	$()$ -- the identity element	1	$\frac{6!}{(1{)}^6(6!)}$	even; no	1	$\mathrm{l}\mathrm{c}\mathrm{m}\{1\}$
2 + 1 + 1 + 1 + 1	2 (1 time), 1 (4 times)	transposition, four fixed points	$(1,2)$	15	$\frac{6!}{(2)(1{)}^4(4!)}$	odd	2	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,1\}$
3 + 1 + 1 + 1	3 (1 time), 1 (3 times)	one 3-cycle, three fixed points	$(1,2,3)$	40	$\frac{6!}{(3)(1{)}^3(3!)}$	even; no	3	$\mathrm{l}\mathrm{c}\mathrm{m}\{3,1\}$
4 + 1 + 1	4 (1 time), 1 (2 times)	one 4-cycle, two fixed points	$(1,2,3,4)$	90	$\frac{6!}{(4)(1{)}^2(2!)}$	odd	4	$\mathrm{l}\mathrm{c}\mathrm{m}\{4,1\}$
2 + 2 + 1 + 1	2 (2 times), 1 (2 times)	double transposition: two 2-cycles, two fixed points	$(1,2)(3,4)$	45	$\frac{6!}{(2{)}^2(2!)(1{)}^2(2!)}$	even; no	2	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,1\}$
5 + 1	5 (1 time), 1 (1 time)	one 5-cycle, one fixed point	$(1,2,3,4,5)$	144	$\frac{6!}{(5)(1)}$	even; yes; yes	5	$\mathrm{l}\mathrm{c}\mathrm{m}\{5,1\}$
3 + 2 + 1	3 (1 time), 2 (1 time), 1 (1 time)	one 3-cycle, one 2-cycle, one fixed point	$(1,2,3)(4,5)$	120	$\frac{6!}{(3)(2)(1)}$	odd	6	$\mathrm{l}\mathrm{c}\mathrm{m}\{3,2,1\}$
2 + 2 + 2	2 (3 times)	triple transposition	$(1,2)(3,4)(5,6)$	15	$\frac{6!}{(2{)}^3(3!)}$	odd	2	$\mathrm{l}\mathrm{c}\mathrm{m}\{2\}$
4 + 2	4 (1 time), 2 (1 time)	one 4-cycle, one 2-cycle	$(1,2,3,4)(5,6)$	90	$\frac{6!}{(4)(2)}$	even; no	4	$\mathrm{l}\mathrm{c}\mathrm{m}\{4,2\}$
3 + 3	3 (2 times)	two 3-cycles	$(1,2,3)(4,5,6)$	40	$\frac{6!}{(3{)}^2(2!)}$	even; no	3	$\mathrm{l}\mathrm{c}\mathrm{m}\{3\}$
6	6 (1 time)	one 6-cycle	$(1,2,3,4,5,6)$	120	$\frac{6!}{6}$	odd	6	$\mathrm{l}\mathrm{c}\mathrm{m}\{6\}$
Total (11 rows = 11 conjugacy classes)	--	--	--	720 (equals order of the whole group)	--	odd: 360, 5 classes even;no: 216, 5 classes even;yes;yes: 144, 1 class	order 1: 1, order 2: 75, order 3: 80, order 4: 180, order 5: 144, order 6: 240	--

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.