Subgroup structure of central product of D8 and Z4
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This article gives specific information, namely, subgroup structure, about a particular group, namely: central product of D8 and Z4.
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The central product of the dihedral group of order eight and cyclic group of order four is a central product of these groups, over a common central subgroup of order two.
It is given by the presentation:
.
Here, is the dihedral group of order eight and
is the cyclic group of order four.
Here is a quick description of all the subgroups of this group:
- The trivial subgroup. Isomorphic to trivial group. (1)
- The subgroup
. This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to cyclic group:Z2. (1)
- The subgroups
,
,
,
. These come in two conjugacy classes of 2-subnormal subgroups, one comprising
and
and the other comprising
and
. However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
- The subgroups
and
. These form a single conjugacy class of 2-subnormal subgroups. Isomorphic to cyclic group:Z2. (2)
- The subgroup
of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
- The subgroups
,
and
. These are normal subgroups but are automorphic subgroups: they are related by outer automorphisms. Isomorphic to cyclic group:Z4. (3)
- The subgroup
,
and
. These are all normal subgroups but are related by outer automorphisms. Isomorphic to Klein four-group. (3)
- The subgroup
. This is an isomorph-free subgroup of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to quaternion group. (3)
- The subgroups
,
and
. These are all normal and related by outer automorphisms. Isomorphic to direct product of Z4 and Z2. (3)
- The subgroups
,
and
. These are all normal and are related by outer automorphisms. Isomorphic to dihedral group:D8. (3)
- The whole group. Isomorphic to central product of D8 and Z4. (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 | ![]() |
7 | 4 | 1 | 1 |
Cyclic group:Z4 | ![]() |
4 | 4 | 4 | 1 |
Klein four-group | ![]() |
3 | 3 | 3 | 0 |
Direct product of Z4 and Z2 | ![]() |
3 | 3 | 3 | 0 |
Dihedral group:D8 | ![]() |
3 | 3 | 3 | 0 |
Quaternion group | ![]() |
1 | 1 | 1 | 1 |
Central product of D8 and Z4 | ![]() |
1 | 1 | 1 | 1 |
Total | -- | 23 | 20 | 17 | 5 |
Table listing number of subgroups by order
Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|
![]() |
1 | 1 | 1 | 1 |
![]() |
7 | 4 | 1 | 1 |
![]() |
7 | 7 | 7 | 1 |
![]() |
7 | 7 | 7 | 1 |
![]() |
1 | 1 | 1 | 1 |
Total | 23 | 20 | 17 | 5 |