Direct product of D8 and Z2

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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This particular group is a finite group of order: 16

Definition

This group is defined as the external direct product of the dihedral group of order eight and the cyclic group of order two.

A presentation for it is:

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

Elements

Upto conjugacy

There are ten conjugacy classes:

  1. The identity element. (1)
  2. The element a^2. (1)
  3. The element y. (1)
  4. The element a^2y . (1)
  5. The two-element conjugacy class comprising a and a^3. (2)
  6. The two-element conjugacy class comprising ay and a^3y. (2)
  7. The two-element conjugacy class comprising x and a^2x. (2)
  8. The two-element conjugacy class comprising xy and a^2xy. (2)
  9. The two-element conjugacy class comprising ax and a^3x. (2)
  10. The two-element conjugacy class comprising axy and a^3xy. (2)

Upto automorphism

Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, y and a^2y are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes 1,1,2,4,8.

Group properties

Property Satisfied Explanation Comment
Abelian group No a and x don't commute
Nilpotent group Yes Prime power order implies nilpotent
Metacyclic group No
Supersolvable group Yes Finite nilpotent implies supersolvable
Solvable group Yes Nilpotent implies solvable
T-group No \langle x \rangle \triangleleft \langle a^2,x \rangle, which is normal, but \langle x \rangle is not normal Smallest example for normality is not transitive.
Monolithic group No Center is Klein four-group, any subgroup of order two is minimal normal.
One-headed group No Distinct maximal subgroups of order eight.
SC-group No
ACIC-group Yes Every automorph-conjugate subgroup is characteristic
Rational group Yes Any two elements that generate the same cyclic group are conjugate Thus, all characters are integer-valued.
Rational-representation group Yes All representations over characteristic zero are realized over the rationals. Contrast with quaternion group, that is rational but not rational-representation.