- Binary octahedral group
In

mathematics , the**binary octahedral group**is an extension of theoctahedral group "O" of order 24 by acyclic group of order 2. It can be defined as thepreimage of the octahedral group under the 2:1covering homomorphism :$mathrm\{Sp\}(1)\; o\; mathrm\{SO\}(3).,$whereSp(1) is the multiplicative group of unitquaternion s. (For a description of this homomorphism see the article onquaternions and spatial rotation s.) It follows that the binary octahedral group isdiscrete subgroup of Sp(1) of order 48.**Elements**Explicitly, the binary octahedral group is given as the union of the 24

Hurwitz unit s:$\{pm\; 1,pm\; i,pm\; j,pm\; k,\; frac\{1\}\{2\}(pm\; 1\; pm\; i\; pm\; j\; pm\; k)\}$with all 24 quaternions obtained from:$frac\{1\}\{sqrt\; 2\}(pm\; 1\; pm\; 1i\; +\; 0j\; +\; 0k)$by apermutation of coordinates (all possible sign combinations). All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).**Properties**The binary octahedral group, denoted by 2"O", fits into the

short exact sequence :$1\; o\{pm\; 1\}\; o\; 2O\; o\; O\; o\; 1.,$This sequence does not split, meaning that 2"O" is "not" asemidirect product of {±1} by "O". In fact, there is no subgroup of 2"O" isomorphic to "O".The center of 2"O" is the subgroup {±1}, so that the

inner automorphism group is isomorphic to "O". The fullautomorphism group is isomorphic to "O" ×**Z**_{2}.**Presentation**The group 2"O" has a presentation given by:$langle\; r,s,t\; mid\; r^2\; =\; s^3\; =\; t^4\; =\; rst\; angle$or equivalently,:$langle\; s,t\; mid\; (st)^2\; =\; s^3\; =\; t^4\; angle.$Generators with these relations are given by:$s\; =\; frac\{1\}\{2\}(1+i+j+k)\; qquad\; t\; =\; frac\{1\}\{sqrt\; 2\}(1+i).$

**ubgroups**The

quaternion group consisting of the 8Lipschitz unit s forms anormal subgroup of 2"O" of index 6. Thequotient group is isomorphic to "S"_{3}(thesymmetric group on 3 letters). Thebinary tetrahedral group , consisting of the 24Hurwitz unit s, forms a normal subgroup of index 2. These two groups, together with the center {±1}, are the only nontrivial normal subgroups of 2"O".The

generalized quaternion group of order 16 also forms a subgroup of 2"O". This subgroup isself-normalizing so itsconjugacy class has 3 members. There are also isomorphic copies of thebinary dihedral group s of orders 8 and 12 in 2"O". All other subgroups arecyclic group s generated by the various elements (with orders 3, 4, 6, and 8).**ee also***

binary polyhedral group

*binary cyclic group

*binary dihedral group

*binary tetrahedral group

*binary icosahedral group **References***cite book | first = John H. | last = Conway | coauthors = Smith, Derek A. | authorlink = John Horton Conway | title = On Quaternions and Octonions | publisher = AK Peters, Ltd | location = Natick, Massachusetts | year = 2003 | isbn = 1-56881-134-9

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