Alternating group:A9
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Contents
Definition
This group is defined as the alternating group of degree 9, i.e., the alternating group on a set of size 9. This can be taken as the group on .
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 181440 | groups with same order | As ![]() ![]() |
exponent of a group | 1260 | groups with same order and exponent of a group | groups with same exponent of a group |
Linear representation theory
Further information: linear representation theory of alternating group:A9
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (such as ![]() ![]() |
1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 maximum: 216, number: 18, sum of squares: 181440 |
minimal splitting field, i.e., smallest field of realization of all irreducible representations (characteristic zero) | ![]() ![]() Same as ![]() Same as field generated by character values |
condition for a field of characteristic not 2,3,5,7 to be a splitting field | -15 should be a square in that field |
GAP implementation
Description | Functions used | Explanation |
---|---|---|
AlternatingGroup(9) | AlternatingGroup |