Direct product of Q8 and Z2

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Definition

This group is the external direct product of the quaternion group of order eight and the cyclic group of order two.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order	The group is an external direct product of a group (quaternion group) of order 8 and a group (cyclic group:Z2) of order 2, and order of direct product is product of orders
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	4	groups with same order and exponent of a group | groups with same exponent of a group	The group is an external direct product of a group (quaternion group) of exponent 4 and a group (cyclic group:Z2) of exponent 2, and exponent of direct product is lcm of exponents
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length	2	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	3	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set	Generating set of quaternion direct factor and generating element for other direct factor (also equal to rank of Frattini quotient).
subgroup rank of a group	3	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group	2	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

Group properties

Property	Satisfied?	Explanation	Comment
Abelian group	No
Nilpotent group	Yes
Group of nilpotency class two	Yes
Metabelian group	Yes
Metacyclic group	No
UL-equivalent group	No	Has nilpotency class two but the derived subgroup (order two) is not equal to the center (order four); also, the group is an external direct product of groups of different nilpotency class values	See also nilpotent not implies UL-equivalent

Subgroups

Further information: Subgroup structure of direct product of Q8 and Z2

The center of the group is isomorphic to the Klein four-group.

The commutator subgroup is isomorphic to cyclic group:Z2.

GAP implementation

Group ID

This finite group has order 16 and has ID 12 among the groups of order 16 in GAP's SmallGroup library. For context, there are groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,12)

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

gap> G := SmallGroup(16,12);

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

IdGroup(G) = [16,12]

or just do:

IdGroup(G)

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