# Alternating group:A9

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## 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 $\{ 1,2,3,4,5,6,7,8,9 \}$.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 181440 groups with same order As $A_n, n = 9$: $n!/2 = 9!/2 = (9 \cdot 8 \cdot 7 \cdot 6 \cdot \dots \cdot 2 \cdot 1)/2 = 181440$
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 $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 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) $\mathbb{Q}(\zeta + \zeta^2 + \zeta^4 + \zeta^8)$ where $\zeta$ is a primitive fifteenth root of unity
Same as $\mathbb{Q}(\sqrt{-15})$
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