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:

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

The group has the following subgroups:

  1. The trivial group. (1)
  2. The cyclic group \langle a^2 \rangle of order two. This equals the commutator subgroup, is central, and is also the set of squares. Isomorphic to cyclic group:Z2. (1)
  3. The subgroups \langle y \rangle and \langle a^2y \rangle. These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
  4. The subgroups \langle x \rangle, \langle ax \rangle, \langle a^2x \rangle, \langle a^3x \rangle, \langle xy \rangle, \langle axy \rangle, \langle a^2xy, and \langle a^3xy \rangle. These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class \langle x \rangle, \langle a^2x \rangle, the class \langle ax \rangle, \langle a^3x \rangle, the class \langle xy \rangle, \langle a^2xy, and the class \langle axy \rangle, \langle a^3xy \rangle. Isomorphic to cyclic group:Z2. (8)
  5. The subgroup \langle a^2, y \rangle. This is the center, hence is a characteristic subgroup. Isomorphic to Klein four-group. (1)
  6. The subgroups \langle a^2, x \rangle, \langle a^2, ax \rangle, \langle a^2, xy \rangle, and \langle a^2, axy \rangle. These are all normal subgroups, but are related by outer automorphisms. Isomorphic to Klein four-group. (4)
  7. The subgroups \langle y, x \rangle, \langle y, ax \rangle, \langle y, a^2x \rangle, \langle y, a^3x, \langle a^2y, x \rangle, \langle a^2y, ax, \langle a^2y, a^2x \rangle, \langle a^2y, a^3x \rangle. 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)
  8. The subgroups \langle a \rangle and \langle ay \rangle. They are both normal and are related via an outer automorphism. Isomorphic to cyclic group:Z4. (2)
  9. The subgroups \langle a^2,x,y \rangle and \langle a^2, ax, y \rangle. These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
  10. The subgroups \langle a,x \rangle, \langle a, xy \rangle, \langle ay, x \rangle and \langle ay, xy \rangle. These are all normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (4)
  11. The subgroup \langle a, y \rangle. This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
  12. 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 1 1
Cyclic group:Z2 (2,1) 11 7 3 1
Cyclic group:Z4 (4,1) 2 2 2 0
Klein four-group (4,2) 13 9 5 1
Direct product of Z4 and Z2 (8,2) 1 1 1 1
Dihedral group:D8| (8,3) 4 4 4 0
Elementary abelian group of order eight (8,5) 2 2 2 0
Direct product of D8 and Z2 (16,11) 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 1
2 11 7 3 1
4 15 11 7 1
8 7 7 7 1
16 1 1 1 1
Total 35 27 19 5

The commutator subgroup (type (2))

This is the two-element subgroup generated by a^2.

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:

It also satisfies the following properties:

Subgroup properties not satisfied by this subgroup

The center (type (5))

This is a Klein four-subgroup comprising the identity, a^2, y and a^2y.

Subgroup-defining functions yielding this subgroup

Subgroup properties satisfied by this subgroup

Subgroup properties not satisfied by this subgroup

The unique characteristic subgroup of order eight (type (10))

This is a subgroup generated by a and y, and is the direct product of a cyclic group of order four generated by a and a cyclic group of order two generated by y.

Subgroup-defining functions yielding this subgroup

Subgroup properties satisfied by this subgroup

Subgroup properties not satisfied by this subgroup