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Dicyclic group:Dic20

Revision as of 19:04, 17 September 2009 by Vipul (talk | contribs) (Definition)

Definition

This group is defined as the dicyclic group of order 20, and hence degree 5. In other words, it has the presentation:

\langle a,b,c \mid a^5 = b^2 = c^2 = abc \rangle

Alternatively, it has the presentation:

\langle a,b,c \mid a^{10} = b^4 = e, bab^{-1} = a^{-1} \rangle.

Arithmetic functions

Function Value Explanation
order 20
exponent 10
Frattini length 2
derived length 2
nilpotency class -- Not a nilpotent group.
minimum size of generating set 2
subgroup rank 2

Group properties

GAP implementation

Group ID

This finite group has order 20 and has ID 1 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(20,1)

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

gap> G := SmallGroup(20,1);

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

IdGroup(G) = [20,1]

or just do:

IdGroup(G)

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