Symmetric group:S6
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Contents
Definition
The symmetric group , 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. In particular, it is a symmetric group on finite set.
- It is the symplectic group , and hence also the projective symplectic group (see isomorphism between symplectic and projective symplectic group in characteristic two).
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 720#Arithmetic functions
Basic arithmetic functions
Arithmetic functions of a counting nature
Function | Value | Similar groups | Explanation for function value |
---|---|---|---|
number of conjugacy classes | 11 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | As the number of conjugacy classes is , where is the number of unordered integer partitions; see cycle type determines conjugacy class. As (even): . More here See element structure of symmetric group:S6 |
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 | |
number of subgroups | 1455 | groups with same order and number of subgroups | groups with same number of subgroups |
Group properties
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No | and don't commute. In fact, is non-abelian for . |
nilpotent group | No | |
solvable group | No | |
simple group | No | has alternating group:A6 as a proper nontrivial normal subgroup |
almost simple group | Yes | sandwiched between the simple group alternating group:A6 and the automorphism group thereof. See also symmetric groups are almost simple. |
quasisimple group | No | |
one-headed group | Yes | alternating group:A6 is the unique maximal normal subgroup |
monolithic group | Yes | alternating group:A6 is the unique minimal normal subgroup |
T-group | Yes | |
rational representation group | Yes | See symmetric groups are rational representation, linear representation theory of symmetric groups |
rational group | Yes | See symmetric groups are rational |
N-group | Yes | See classification of symmetric groups that are N-groups |
Elements
Further information: element structure of symmetric group:S6
Up to conjugacy
For convenience, we take the underlying set here as .
There are eleven conjugacy classes, corresponding to the unordered integer partitions of (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 | even; no | 1 | ||
2 + 1 + 1 + 1 + 1 | 2 (1 time), 1 (4 times) | transposition, four fixed points | 15 | odd | 2 | |||
3 + 1 + 1 + 1 | 3 (1 time), 1 (3 times) | one 3-cycle, three fixed points | 40 | even; no | 3 | |||
4 + 1 + 1 | 4 (1 time), 1 (2 times) | one 4-cycle, two fixed points | 90 | odd | 4 | |||
2 + 2 + 1 + 1 | 2 (2 times), 1 (2 times) | double transposition: two 2-cycles, two fixed points | 45 | even; no | 2 | |||
5 + 1 | 5 (1 time), 1 (1 time) | one 5-cycle, one fixed point | 144 | even; yes; yes | 5 | |||
3 + 2 + 1 | 3 (1 time), 2 (1 time), 1 (1 time) | one 3-cycle, one 2-cycle, one fixed point | 120 | odd | 6 | |||
2 + 2 + 2 | 2 (3 times) | triple transposition | 15 | odd | 2 | |||
4 + 2 | 4 (1 time), 2 (1 time) | one 4-cycle, one 2-cycle | 90 | even; no | 4 | |||
3 + 3 | 3 (2 times) | two 3-cycles | 40 | even; no | 3 | |||
6 | 6 (1 time) | one 6-cycle | 120 | odd | 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 | , | 15 | 2 | 30 | 2 |
3 + 1 + 1 + 1, 3 + 3 | , | 40 | 2 | 80 | 3 |
4 + 1 + 1 | 90 | 1 | 90 | 4 | |
4 + 2 | 90 | 1 | 90 | 4 | |
5 + 1 | 144 | 1 | 144 | 5 | |
3 + 2 + 1, 6 | , | 120 | 2 | 240 | 6 |
2 + 2 + 1 + 1 | 45 | 1 | 45 | 2 | |
Total (8 classes) | -- | -- | 11 | 720 | -- |
Subgroups
Further information: subgroup structure of symmetric group:S6
Quick summary
Item | Value |
---|---|
Number of subgroups | 1455 Compared with : 1, 2, 6, 30, 156, 1455, 11300, 151221, ... |
Number of conjugacy classes of subgroups | 56 Compared with : 1, 2, 4, 11, 19, 56, 96, 296, ... |
Number of automorphism classes of subgroups | 37 Compared with : 1, 2, 4, 11, 19, 37, 96, 296, ... |
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems | 2-Sylow: direct product of D8 and Z2 (order 16), Sylow number is 45 3-Sylow: elementary abelian group:E9 (order 9), Sylow number is 10 5-Sylow: cyclic group:Z5 (order 5), Sylow number is 36 |
Hall subgroups | No Hall subgroups other than the Sylow subgroups, whole group, and trivial subgroup. In particular, there is no -Hall subgroup, -Hall subgroup, and -Hall subgroup. |
maximal subgroups | maximal subgroups have order 48, 72, 120, and 360 |
normal subgroups | The only normal subgroups are the whole group, the trivial subgroup, and alternating group:A6 as A6 in S6. |
Linear representation theory
Further information: linear representation theory of symmetric group:S6
Summary
Item | Summary |
---|---|
Degrees of irreducible representations over a splitting field | 1,1,5,5,5,5,9,9,10,10,16 maximum: 16, lcm: 720, number: 11, sum of squares: 720 |
Schur index values of irreducible representations | 1,1,1,1,1,1,1,1,1,1,1 |
Smallest ring of realization for all irreducible representations (characteristic zero) | -- ring of integers |
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) | -- hence it is a rational representation group |
Criterion for a field to be a splitting field | Any field of characteristic not 2,3, or 5 |
Smallest size splitting field | field:F7 |
Character table
The character table below is incomplete, it contains only eight of the eleven representations.
Representation/conjugacy class representative and size | (size 1) | (size 15) | (size 15) | (size 40) | (size 40) | (size 45) | (size 90) | (size 90) | (size 120) | (size 120) | -- size 144 |
---|---|---|---|---|---|---|---|---|---|---|---|
trivial | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
sign | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 |
standard | 5 | 3 | -1 | 2 | -1 | 1 | 1 | -1 | 0 | -1 | 0 |
product of standard and sign | 5 | -3 | 1 | 2 | -1 | 1 | -1 | -1 | 0 | 1 | 0 |
other five-dimensional irreducible | 5 | -1 | 3 | -1 | 2 | 1 | 1 | -1 | -1 | 0 | 0 |
other five-dimensional irreducible | 5 | 1 | -3 | -1 | 2 | 1 | -1 | -1 | 1 | 0 | 0 |
second exterior power of standard | 10 | 2 | -2 | 1 | 1 | -2 | 0 | 0 | -1 | 1 | 0 |
third exterior power of standard | 10 | -2 | 2 | 1 | 1 | -2 | 0 | 0 | 1 | -1 | 0 |
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 840 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 :
gap> G := SmallGroup(720,763);
Conversely, to check whether a given group 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.
Other descriptions
Description | Functions used |
---|---|
SymmetricGroup(6) | SymmetricGroup |
Sp(4,2) | Sp |
PSp(4,2) | PSp |