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> |
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− | | {{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 | ![]() |
1 | ![]() |
even;no | 1 |
2 + 1 + 1 + 1 + 1 + 1 + 1 | transposition, six fixed points | ![]() |
28 | ![]() ![]() |
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 |