Direct product of Z9 and E27

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Definition

The group is defined in the following equivalent ways:

1. It is the external direct product of the cyclic group of order 9 and the elementary abelian group of order 27, i.e., $\Z_9 \times E_{27}$.
2. it is the external direct product of one copy of the cyclic group of order 9 and three copies of the cyclic group of order 3.

Arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 243 groups with same order Since order of direct product is product of orders, the order is $9 \times 27 = 243$
prime-base logarithm of order 5
exponent 9 groups with same order and exponent | groups with same exponent Exponent of direct product is lcm of exponents. Thus, the exponent is $\operatorname{lcm} \{ 9,3 \} = 9$
prime-base logarithm of exponent 2
minimum size of generating set 4 groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 4 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 4 groups with same order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 4 groups with same order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 4 groups with same order and characteristic rank of a p-group | groups with same characteristic rank of a p-group
derived length 1 groups with same order and derived length | groups with same derived length
nilpotency class 1 groups with same order and nilpotency class | groups with same nilpotency class
Template:Arithmetic function vlaue given order

GAP implementation

Group ID

This finite group has order 243 and has ID 61 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(243,61)

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

gap> G := SmallGroup(243,61);

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

IdGroup(G) = [243,61]

or just do:

IdGroup(G)

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

Short descriptions

Description GAP functions used Mathematical translation of description
DirectProduct(CyclicGroup(9),ElementaryAbelianGroup(27)) CyclicGroup, DirectProduct, ElementaryAbelianGroup external direct product of cyclic group of order 9 and elementary abelian group of order 27
DirectProduct(CyclicGroup(9),CyclicGroup(3),CyclicGroup(3),CyclicGroup(3)) CyclicGroup, DirectProduct external direct product of one copy of cyclic group of order 9 and three copies of cyclic group of order 3