Nontrivial semidirect product of Z7 and Z9

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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:


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:


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 ]