Difference between revisions of "Symmetric group:S8"
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| {{arithmetic function value order|40320}}|| The order is <math>8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1</math> | | {{arithmetic function value order|40320}}|| The order is <math>8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1</math> | ||
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| − | | {{arithmetic function value given order|exponent of a group|840}}|| The exponent is the least common multiple of <math>1,2,3,4,5,6,7,8</math> | + | | {{arithmetic function value given order|exponent of a group|840|40320}}|| The exponent is the least common multiple of <math>1,2,3,4,5,6,7,8</math> |
|- | |- | ||
| − | | {{arithmetic function value given order|Frattini length|1}}|| | + | | {{arithmetic function value given order|Frattini length|1|40320}}|| |
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Revision as of 17:11, 28 April 2012
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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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 power degree.
Arithmetic functions
| Function | Value | Similar groups | Explanation |
|---|---|---|---|
| order (number of elements, equivalently, cardinality or size of underlying set) | 40320 | groups with same order | The order is 
|
| exponent of a group | 840 | 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:S8
Upto 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 + 1 | all points fixed |  -- the identity element |
1 |  |
even;no | 1 |
| 2 + 1 + 1 + 1 + 1 + 1 + 1 | transposition, six fixed points |  |
28 |  , also  |
odd | 2 |
| 3 + 1 + 1 + 1 + 1 + 1 | one 3-cycle, five fixed points |  |
112 |  |
even;no | 3 |
| 4 + 1 + 1 + 1 + 1 | one 4-cycle, four fixed points |  |
420 |  |
odd | 4 |
| 2 + 2 + 1 + 1 + 1 + 1 | two transpositions, four fixed points |  |
210 |  |
even;no | 2 |
| 5 + 1 + 1 + 1 | one 5-cycle, three fixed points |  |
1344 |  |
even;no | 5 |
| 3 + 2 + 1 + 1 + 1 | one 3-cycle, one transposition, three fixed points |  |
1120 |  |
odd | 6 |
| 6 + 1 + 1 | one 6-cycle, two fixed points |  |
3360 |  |
odd | 6 |
| 4 + 2 + 1 + 1 | one 4-cycle, one 2-cycle, two fixed points |  |
2520 |  |
even;no | 4 |
| 2 + 2 + 2 + 1 + 1 | three 2-cycles, two fixed points |  |
420 |  |
odd | 2 |
| 3 + 3 + 1 + 1 | two 3-cycles, two fixed points |  |
1120 |  |
even;no | 3 |
| 7 + 1 | one 7-cycle, one fixed point |  |
5760 |  |
even;yes;no | 7 |
| 3 + 2 + 2 + 1 | one 3-cycle, two transpositions, one fixed point |  |
1680 |  |
even;no | 6 |
| 4 + 3 + 1 | one 4-cycle, one 3-cycle, one fixed point |  |
3360 |  |
odd | 12 |
| 5 + 2 + 1 | one 5-cycle, one 2-cycle, one fixed point |  |
4032 |  |
odd | 10 |
| 2 + 2 + 2 + 2 | four 2-cycles |  |
105 |  |
even;no | 2 |
| 4 + 2 + 2 | one 4-cycle, two 2-cycles |  |
1260 |  |
odd | 4 |
| 3 + 3 + 2 | two 3-cycles, one 2-cycle |  |
1120 |  |
odd | 6 |
| 6 + 2 | one 6-cycle, one 2-cycle |  |
3360 |  |
even;no | 6 |
| 5 + 3 | one 5-cycle, one 3-cycle |  |
2688 |  |
even;yes;no | 15 |
| 4 + 4 | two 4-cycles |  |
1260 |  |
even;no | 4 |
| 8 | one 8-cycle |  |
5040 |  |
odd | 8 |
GAP implementation
| Description | Functions used |
|---|---|
| SymmetricGroup(8) | SymmetricGroup |
