octic group

The octic group also known as the 4th dihedral groupMathworldPlanetmath, is a non-Abelian groupMathworldPlanetmath with eight elements. It is traditionally denoted by D4. This group is defined by the presentationMathworldPlanetmathPlanetmath


or, equivalently, defined by the multiplication table

e s s2 s3 t ts ts2 ts3
e e s s2 s3 t ts ts2 ts3
s s s2 s3 e ts3 t ts ts2
s2 s2 s3 e s ts2 ts3 t ts
s3 s3 e s s2 ts ts2 ts3 t
t t ts ts2 ts3 e s s2 s3
ts ts ts2 ts3 t s3 e s s2
ts2 ts2 ts3 t ts s2 s3 e s
ts3 ts3 t ts ts2 s s2 s3 e

where we have put each productPlanetmathPlanetmath xy into row x and column y. The lattice of the subgroupsMathworldPlanetmathPlanetmath is given below:

where <a1,,an> denotes the subgroup generated by a1,,an and {b1,,bn} denotes the subgroup. Of those subgroups, the following are its proper normal subgroupMathworldPlanetmath: {e,s2,t,ts2}, <s>, {e,s2,st,ts}, and <s2>. In addition the center and commutator subgroupMathworldPlanetmath of the octic group is <s2>. It can also be shown that the automorphismPlanetmathPlanetmathPlanetmathPlanetmath of the octic group (Aut(D4)) is isomorphic to itself(D4).[PJ] An additional property is that the subgroup of the general linear groupMathworldPlanetmath of dimensionMathworldPlanetmath 2 over the real numbers generated by: [ 0 1 -1 0 ],[ 0 1 1 0 ] is isomorphic to the octic group.


  • PJ Pedersen, John: Groups of small order. http://www.math.usf.edu/ eclark/algctlg/small_groups.htmlhttp://www.math.usf.edu/ eclark/algctlg/small_groups.html
Title octic group
Canonical name OcticGroup
Date of creation 2013-03-22 14:47:59
Last modified on 2013-03-22 14:47:59
Owner Daume (40)
Last modified by Daume (40)
Numerical id 5
Author Daume (40)
Entry type Example
Classification msc 20F55
Synonym D4