Symmetric group:S8
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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 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, two 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 |
Subgroups
Further information: subgroup structure of symmetric group:S8
Quick summary
Item | Value |
---|---|
Number of subgroups | 151221 Compared with : 1,2,6,30,156,1455,11300,151221 |
Number of conjugacy classes of subgroups | 296 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,... |
Linear representation theory
Further information: linear representation theory of symmetric group:S8
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1,1,7,7,14,14,20,20,21,21,28,28,35,35,42,56,56,64,64,70,70,90 maximum: 90, lcm: 20160, number: 22, sum of squares: 40320 |
Schur index values of irreducible representations over a splitting field | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 |
smallest ring of realization (characteristic zero) | -- ring of integers |
smallest field of realization (characteristic zero), i.e., smallest splitting field in characteristic zero | -- hence it is a rational-representation group |
condition for a field to be a splitting field | any field of characteristic not 2,3,5,7 |
smallest size finite splitting field | field:F11 |
GAP implementation
Description | Functions used |
---|---|
SymmetricGroup(8) | SymmetricGroup |