# Double cover of symmetric group:S5 of minus type

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## Definition

This group is defined as the double cover of symmetric group for symmetric group:S5 that is of "-" type. In particular it is one of the Schur covering groups for symmetric group:S5, and is thus one of the stem extensions where the base normal subgroup is cyclic group:Z2 and the quotient group is symmetric group:S5.

The group is denoted $2 \cdot S_5^-$.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 240#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 240 groups with same order order of extension group is product of order of normal subgroup and quotient group: the order is $2(n!) = 2(5!) = 2(120) = 240$.

## GAP implementation

### Group ID

This finite group has order 240 and has ID 89 among the groups of order 240 in GAP's SmallGroup library. For context, there are 208 groups of order 240. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(240,89)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(240,89);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [240,89]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.