SmallGroup(16,3)
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Contents
Definition
This group is a semidirect product , and can be defined using the following presentation:
Note that is generated by
alone, because the final relation allows us to write
in terms of
and
.
The subgroup is isomorphic to the direct product of Z4 and Z2, and the element
is an element of order two that acts on the subgroup
by conjugation by fixing
and sending
to
.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Group properties
Want to compare with other groups of the same order? Check out groups of order 16#Group properties.
Property | Satisfied | Explanation |
---|---|---|
abelian group | No | ![]() |
group of prime power order | Yes | |
nilpotent group | Yes | prime power order implies nilpotent |
group of nilpotency class two | Yes | |
supersolvable group | Yes | |
T-group | No | The subgroup ![]() |
monolithic group | No | |
one-headed group | No | |
ambivalent group | No | |
rational group | No | |
rational-representation group | No |
Subgroups
Further information: subgroup structure of SmallGroup(16,3)
- The trivial group. (1)
- The subgroup
, which is the unique characteristic subgroup of order
. Isomorphic to cyclic group:Z2. It is the commutator subgroup, and can also be described as the unique group of order two containing an element that is not a square but is a product of squares. The quotient group is isomorphic to direct product of Z4 and Z2. (1)
- The subgroups
and
, which are both normal subgroups related by an outer automorphism. Isomorphic to cyclic group:Z2. The quotient group for each is isomorphic to dihedral group:D8. (2)
- The subgroups
,
,
and
. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to cyclic group:Z2. (4)
- The subgroup
, which is the center, first agemo subgroup, and Frattini subgroup. Isomorphic to Klein four-group. The quotient group is isomorphic to Klein four-group. (1)
- The subgroups
and
. Both are normal subgroups and are related by an outer automorphism. Isomorphic to Klein four-group. The quotient group is isomorphic to cyclic group:Z4. (2)
- The subgroups
,
,
, and
. Two conjugacy classes of size two each. Isomorphic to Klein four-group. The quotient group is also isomorphic to a Klein four-group. (4)
- The subgroups
,
,
, and
. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each:
is conjugate to
, while
is conjugate to
. Isomorphic to cyclic group:Z4. (4)
- The subgroups
and
. These are both normal subgroups related by an outer automorphism. Isomorphic to direct product of Z4 and Z2. The quotient group is isomorphic to cyclic group:Z2. (2)
- The subgroup
. Isomorphic to elementary abelian group:E8. The quotient group is isomorphic to cyclic group:Z2. (1)
- The whole group. (1)
Subgroup-defining functions
Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|
Center | (4) | center of SmallGroup(16,3) | Klein four-group | |
Commutator subgroup | (2) | commutator subgroup of SmallGroup(16,3) | cyclic group:Z2 | |
Frattini subgroup | (4) | center of SmallGroup(16,3) | Klein four-group | |
Socle | (4) | center of SmallGroup(16,3) | Klein four-group | |
first omega subgroup | (7) | elementary abelian group:E8 |
GAP implementation
Group ID
This finite group has order 16 and has ID 3 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(16,3)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(16,3);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [16,3]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
The group can be constructed using the following GAP commands:
gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^4, F.2^2, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^2,F.3*F.2*F.3^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.2^(-1)*F.1^(-1)]; <fp group on the generators [ f1, f2, f3 ]>