Symmetric group:S7
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Contents
Definition
This group is a finite group defined as the symmetric group on a set of size . The set is typically taken to be .
In particular, it is a symmetric group on finite set as well as a symmetric group of prime degree.
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 5040 | groups with same order | The order is |
exponent of a group | 420 | groups with same order and exponent of a group | groups with same exponent of a group | The exponent is the least common multiple of |
Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length |
Elements
Further information: element structure of symmetric group:S7
Up to conjugacy
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 + 1 | seven fixed points | -- the identity element | 1 | even; no | 1 | |
2 + 1 + 1 + 1 + 1 + 1 | transposition, five fixed points | 21 | , also in this case | odd | 2 | |
3 + 1 + 1 + 1 + 1 | one 3-cycle, four fixed points | 70 | even; no | 3 | ||
4 + 1 + 1 + 1 | one 4-cycle, three fixed points | 210 | odd | 4 | ||
2 + 2 + 1 + 1 + 1 | two 2-cycles, three fixed points | 105 | even;no | 2 | ||
5 + 1 + 1 | one 5-cycle, two fixed points | 504 | even; no | 5 | ||
3 + 2 + 1 + 1 | one 3-cycle, one 2-cycle, two fixed points | 420 | odd | 6 | ||
6 + 1 | one 6-cycle, one fixed point | 840 | odd | 6 | ||
4 + 2 + 1 | one 4-cycle, one 2-cycle, one fixed point | 630 | even;no | 4 | ||
2 + 2 + 2 + 1 | three 2-cycles, one fixed point | 105 | odd | 2 | ||
3 + 3 + 1 | two 3-cycles, one fixed point | 280 | even;no | 3 | ||
3 + 2 + 2 | one 3-cycle, two transpositions | 210 | even;no | 6 | ||
5 + 2 | one 5-cycle, one transposition | 504 | odd | 10 | ||
4 + 3 | one 4-cycle, one 3-cycle | 420 | odd | 12 | ||
7 | one 7-cycle | 720 | even;yes;no | 7 |
Subgroups
Further information: subgroup structure of symmetric group:S7
Quick summary
Item | Value |
---|---|
Number of subgroups | 11300 Compared with : 1,2,6,30,156,1455,11300,151221 |
Number of conjugacy classes of subgroups | 96 Compared with : 1,2,4,11,19,56,96,296,554,1593,... |
Number of automorphism classes of subgroups | 96 Compared with : 1,2,4,11,19,37,96,296,554,1593,... |
Isomorphism classes of Sylow subgroups | 2-Sylow: direct product of D8 and Z2 (order 16) 3-Sylow: elementary abelian group:E9 (order 9) 5-Sylow: cyclic group:Z5 (order 5) 7-Sylow: cyclic group:Z7 (order 7) |
Hall subgroups | Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups (of order 144) and -Hall subgroups (of order 720), the latter being S6 in S7. Note that the -Hall subgroups are not contained in -Hall subgroups. |
maximal subgroups | maximal subgroups have orders 42, 144, 240, 720, 2520 |
normal subgroups | the whole group, trivial subgroup, and alternating group:A7 (embedded as A7 in S7) |
subgroups that are simple non-abelian groups | alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size ), alternating group:A6 (order 360), alternating group:A7 (order 2520) |
Linear representation theory
Further information: linear representation theory of symmetric group:S7
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1,1,6,6,14,14,14,14,15,15,20,21,21,35,35 maximum: 35, lcm: 420, number: 15, sum of squares: 5040 |
Schur index values of irreducible representations over a splitting field | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s) |
smallest ring of realization (characteristic zero) | -- ring of integers |
smallest field of realization (characteristic zero) | -- field of rational numbers |
condition for a field to be a splitting field | Any field of characteristic not 2, 3, 5, or 7. |
smallest size splitting field | field:F11, i.e., the field with 11 elements |
GAP implementation
Description | Functions used |
---|---|
SymmetricGroup(7) | SymmetricGroup |