Nontrivial semidirect product of Z7 and Z9
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Contents
Definition
This group is defined as the external semidirect product of cyclic group:Z7 (the normal subgroup, which is being acted upon) and cyclic group:Z9 (the acting group) where the generator of the latter acts by an automorphism of order three. An explicit presentation (where is the identity element) is below:
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 63#Arithmetic functions
Function | Value | Similar groups | Explanation for function value | GAP verification (set G := SmallGroup(63,1);) |
---|---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 63 | groups with same order | order of semidirect product is product of orders: order is . | Order(G); using Order. |
exponent of a group | 63 | groups with same order and exponent of a group | groups with same exponent of a group | There are elements of order 7 and 9, so the exponent is at least their lcm, which is 63. Also, exponent divides order, so the exponent is at most 63. | Exponent(G); using Exponent. |
derived length | 2 | groups with same order and derived length | groups with same derived length | The normal subgroup is an abelian normal subgroup and the quotient is abelian, so the group is a metabelian group. | DerivedLength(G); using DerivedLength. |
Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | FrattiniLength(G); using FrattiniLength. | |
Fitting length | 2 | groups with same order and Fitting length | groups with same Fitting length | The Fitting subgroup is isomorphic to cyclic group:Z21 (direct product of cyclic groups of order 7 and 3), generated by and . | |
minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | The group is not cyclic, but has a generating set of size two, as illustrated by the presentation. | |
subgroup rank | 2 | groups with same order and subgroup rank | groups with same subgroup rank | All subgroups are either cyclic or have generating sets of size two. In fact, all subgroups are metacyclic groups |
Distinguishing features
Smallest of its kind
This group is the smallest odd-order group with an irreducible representation whose Schur index is strictly greater than 1 (the value of the Schur index is 3).
GAP implementation
Group ID
This finite group has order 63 and has ID 1 among the groups of order 63 in GAP's SmallGroup library. For context, there are groups of order 63. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(63,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(63,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [63,1]
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(2);; gap> G := F/[F.1^7,F.2^9,F.2*F.1*F.2^(-1)*F.1^(-2)]; <fp group on the generators [ f1, f2 ]> gap> IdGroup(G); [ 63, 1 ]