Subgroup structure of direct product of D8 and Z2
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This article gives specific information, namely, subgroup structure, about a particular group, namely: direct product of D8 and Z2.
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This article discusses the subgroup structure of the direct product of D8 and Z2.
A presentation for the group that we use is:
.
The group has the following subgroups:
- The trivial group. (1)
- The cyclic group
of order two. This equals the commutator subgroup, is central, and is also the set of squares. Isomorphic to cyclic group:Z2. (1)
- The subgroups
and
. These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
- The subgroups
,
,
,
,
,
,
, and
. These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class
, the class
, the class
, and the class
. Isomorphic to cyclic group:Z2. (8)
- The subgroup
. This is the center, hence is a characteristic subgroup. Isomorphic to Klein four-group. (1)
- The subgroups
,
,
, and
. These are all normal subgroups, but are related by outer automorphisms. Isomorphic to Klein four-group. (4)
- The subgroups
,
,
,
,
,
,
,
. These subgroups are all 2-subnormal subgroups and are related by outer automorphisms, and they come in four conjugacy classes of size two. Isomorphic to Klein four-group. (8)
- The subgroups
and
. They are both normal and are related via an outer automorphism. Isomorphic to cyclic group:Z4. (2)
- The subgroups
and
. These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
- The subgroups
,
,
and
. These are all normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (4)
- The subgroup
. This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
- The whole group. (1)
Tables for quick information
Table classifying isomorphism types of subgroups
Group name | GAP ID | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|
Trivial group | ![]() |
1 | 1 | 1 | 1 |
Cyclic group:Z2 | ![]() |
11 | 7 | 3 | 1 |
Cyclic group:Z4 | ![]() |
2 | 2 | 2 | 0 |
Klein four-group | ![]() |
13 | 9 | 5 | 1 |
Direct product of Z4 and Z2 | ![]() |
1 | 1 | 1 | 1 |
Dihedral group:D8 | ![]() |
4 | 4 | 4 | 0 |
Elementary abelian group of order eight | ![]() |
2 | 2 | 2 | 0 |
Direct product of D8 and Z2 | ![]() |
1 | 1 | 1 | 1 |
Total | -- | 35 | 27 | 19 | 5 |
Table listing number of subgroups by order
Group name | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|
![]() |
1 | 1 | 1 | 1 |
![]() |
11 | 7 | 3 | 1 |
![]() |
15 | 11 | 7 | 1 |
![]() |
7 | 7 | 7 | 1 |
![]() |
1 | 1 | 1 | 1 |
Total | 35 | 27 | 19 | 5 |
The commutator subgroup (type (2))
This is the two-element subgroup generated by .
Subgroup-defining functions yielding this subgroup
Subgroup properties satisfied by this subgroup
On account of being a commutator subgroup as well as an agemo subgroup, this subgroup is a verbal subgroup. Thus, it satisfies the following subgroup properties:
- Fully invariant subgroup
- Image-closed fully invariant subgroup
- Characteristic subgroup
- Image-closed characteristic subgroup
It also satisfies the following properties:
Subgroup properties not satisfied by this subgroup
- Intermediately characteristic subgroup
- Normal-isomorph-free subgroup: There are other normal subgroups isomorphic to it.
- Complemented normal subgroup, lattice-complemented subgroup, permutably complemented subgroup: This follows from the general phenomenon that nilpotent implies center is normality-large.
The center (type (5))
This is a Klein four-subgroup comprising the identity, ,
and
.
Subgroup-defining functions yielding this subgroup
Subgroup properties satisfied by this subgroup
- Characteristic subgroup
- Central subgroup, central factor, transitively normal subgroup
- Characteristic central factor
- Normality-large subgroup: This is more general, since nilpotent implies center is normality-large.
Subgroup properties not satisfied by this subgroup
- Verbal subgroup
- Fully invariant subgroup
- Normal-isomorph-free subgroup
- Lattice-complemented subgroup, permutably complemented subgroup: This follows from the fact that nilpotent implies center is normality-large.
The unique characteristic subgroup of order eight (type (10))
This is a subgroup generated by and
, and is the direct product of a cyclic group of order four generated by
and a cyclic group of order two generated by
.
Subgroup-defining functions yielding this subgroup
- Maximal among abelian characteristic subgroups: It is the unique maximal among abelian characteristic subgroups. In general, there need not be a unique subgroup maximal among abelian characteristic subgroups. Further information: Maximal among abelian characteristic subgroups may be multiple and isomorphic
- The unique constructibly critical subgroup.
Subgroup properties satisfied by this subgroup
- Isomorph-free subgroup
- Intermediately characteristic subgroup
- Maximal characteristic subgroup
- Characteristic subgroup
- Self-centralizing subgroup
- Maximal among abelian characteristic subgroups
- Maximal among abelian normal subgroups
- Abelian critical subgroup, constructibly critical subgroup, critical subgroup