# Nontrivial semidirect product of Z7 and Z9

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## 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 $e$ is the identity element) is below:

$G := \langle a,b \mid a^7 = b^9 = e, bab^{-1} = a^2 \rangle$

## 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 $7 \times 9 =63$. 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 $\langle a \rangle$ 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 $a$ and $b^3$.
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 $G$:

gap> G := SmallGroup(63,1);

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

IdGroup(G) = [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 ]