# Direct product of SL(2,3) and Z2

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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 defined as the external direct product of special linear group:SL(2,3) (the special linear group of degree two over field:F3, the field of three elements) and cyclic group:Z2 (the cyclic group of two elements). It is denoted $SL(2,3) \times \mathbb{Z}_2$ or $SL(2,3) \times C_2$.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 48#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 48 groups with same order order of direct product is product of orders: the order is $24 \times 2$, where $24 = (3^2 - 1)(3^2-3)/2$ is the order of special linear group:SL(2,3) and $2$ is the order of cyclic group:Z2.
exponent of a group 12 groups with same order and exponent of a group | groups with same exponent of a group exponent of direct product is lcm of exponents: the exponent is $\operatorname{lcm}\{ 12,2 \} = 12$.
derived length 3 groups with same order and derived length | groups with same derived length derived length of direct product is maximum of derived lengths: the derived length is $\max \{ 3, 1 \} = 3$.

## GAP implementation

### Group ID

This finite group has order 48 and has ID 32 among the groups of order 48 in GAP's SmallGroup library. For context, there are groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,32)

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

gap> G := SmallGroup(48,32);

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

IdGroup(G) = [48,32]

or just do:

IdGroup(G)

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

### Other descriptions

Description Functions used
DirectProduct(SL(2,3),CyclicGroup(2)) DirectProduct, SL, CyclicGroup