Subgroup structure of central product of D8 and Z4

From Groupprops
Revision as of 14:33, 6 July 2019 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article gives specific information, namely, subgroup structure, about a particular group, namely: central product of D8 and Z4.
View subgroup structure of particular groups | View other specific information about central product of D8 and Z4

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:

G := \langle a,x,y \mid a^4 = x^2 = e, a^2 = y^2, xax = a^{-1}, ay = ya, xy = yx \rangle.

Here, \langle a,x \rangle is the dihedral group of order eight and \langle y \rangle is the cyclic group of order four.

Here is a quick description of all the subgroups of this group:

  1. The trivial subgroup. Isomorphic to trivial group. (1)
  2. The subgroup \langle a^2 \rangle. This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to cyclic group:Z2. (1)
  3. The subgroups \langle x \rangle, \langle ax \rangle, \langle a^2x \rangle, \langle a^3x \rangle. These come in two conjugacy classes of 2-subnormal subgroups, one comprising \langle x \rangle and \langle a^2x \rangle and the other comprising \langle ax \rangle and \langle a^3x \rangle. However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
  4. The subgroups \langle ay \rangle and \langle a^3y \rangle. These form a single conjugacy class of 2-subnormal subgroups. Isomorphic to cyclic group:Z2. (2)
  5. The subgroup \langle y \rangle of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
  6. The subgroups \langle a \rangle, \langle xy \rangle and \langle axy \rangle. These are normal subgroups but are automorphic subgroups: they are related by outer automorphisms. Isomorphic to cyclic group:Z4. (3)
  7. The subgroup \langle a^2, x \rangle, \langle a^2, ax \rangle and \langle a^2, ay \rangle. These are all normal subgroups but are related by outer automorphisms. Isomorphic to Klein four-group. (3)
  8. The subgroup \langle a, xy \rangle. This is an isomorph-free subgroup of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to quaternion group. (1)
  9. The subgroups \langle a,y \rangle, \langle x, y \rangle and \langle ax, y \rangle. These are all normal and related by outer automorphisms. Isomorphic to direct product of Z4 and Z2. (3)
  10. The subgroups \langle a,x \rangle, \langle xy, ay \rangle and \langle axy, ay \rangle. These are all normal and are related by outer automorphisms. Isomorphic to dihedral group:D8. (3)
  11. 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 1 1
Cyclic group:Z2 (2,1) 7 4 1 1
Cyclic group:Z4 (4,1) 4 4 4 1
Klein four-group (4,2) 3 3 3 0
Direct product of Z4 and Z2 (8,2) 3 3 3 0
Dihedral group:D8 (8,3) 3 3 3 0
Quaternion group (8,4) 1 1 1 1
Central product of D8 and Z4 (16,13) 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 1
2 7 4 1 1
4 7 7 7 1
8 7 7 7 1
16 1 1 1 1
Total 23 20 17 5