SmallGroup(128,1015)

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Definition

This is a group of order 128 given by the following presentation, where e is used to denote the identity element:

G := \langle a_1, a_2, a_3, a_4 \mid a_1^2 = a_2^4 = a_3^4 = a_4^4, [a_1,a_2] = a_3^2, [a_1,a_3] = e, [a_2,a_3] = a_4^2, [a_1,a_4] = [a_2,a_4] = [a_3,a_4] = e \rangle

For the definition, [ , ] stands for the commutator. It does not matter whether we choose the left or right action convention for the commutator -- the groups defined in both cases are isomorphic.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 128#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) 128 groups with same order
prime-base logarithm of order 7 groups with same prime-base logarithm of order
max-length of a group 7 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 7 chief length equals prime-base logarithm of order for group of prime power order
composition length 7 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
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 4 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
rank of a p-group 4 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 4 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 4 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

GAP implementation

Group ID

This finite group has order 128 and has ID 1015 among the groups of order 128 in GAP's SmallGroup library. For context, there are groups of order 128. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(128,1015)

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

gap> G := SmallGroup(128,1015);

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

IdGroup(G) = [128,1015]

or just do:

IdGroup(G)

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


Description by presentation

gap> F := FreeGroup(4);
<free group on the generators [ f1, f2, f3, f4 ]>
gap> G := F/[F.1^2,F.2^4,F.3^4,F.4^4,Comm(F.1,F.2)*F.3^(-2),Comm(F.1,F.3),Comm(F.2,F.3)*F.4^(-2),Comm(F.1,F.4),Comm(F.2,F.4),Comm(F.3,F.4)];
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> IdGroup(G);
[ 128, 1015 ]