# Binary tetrahedral group

In

mathematics , the**binary tetrahedral group**is an extension of thetetrahedral group "T" of order 12 by acyclic group of order 2.It is the

binary polyhedral group corresponding to the tetrahedral group, and as such can be defined as thepreimage of the tetrahedral 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 tetrahedral group isdiscrete subgroup of Sp(1) of order 24.**Elements**Explicitly, the binary tetrahedral group is given as the

group of units in the ring ofHurwitz integer s. There are 24 such units given by:$\{pm\; 1,pm\; i,pm\; j,pm\; k,\; frac\{1\}\{2\}(pm\; 1\; pm\; i\; pm\; j\; pm\; k)\}$with all possible sign combinations.All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The

convex hull of these 24 elements in 4-dimensional space form aconvex regular 4-polytope called the24-cell .**Properties**The binary tetrahedral group, denoted by 2"T", fits into the

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

inner automorphism group is isomorphic to "T". The fullautomorphism group is isomorphic to "S"_{4}(thesymmetric group on 4 letters).The binary tetrahedral group can be written as a

semidirect product :$2T=Q\; timesmathbb\; Z\_3$where "Q" is thequaternion group consisting of the 8Lipschitz unit s and**Z**_{3}is thecyclic group of order 3 generated by ω = −½(1+"i"+"j"+"k"). The group**Z**_{3}acts on the normal subgroup "Q" by conjugation. Conjugation by ω is the automorphism of "Q" that cyclically rotates "i", "j", and "k".One can show that the binary tetrahedral group is isomorphic to the

special linear group SL(2,3) — the group of all 2×2 matrices over thefinite field **F**_{3}with unit determinant.**Presentation**The group 2"T" has a presentation given by:$langle\; r,s,t\; mid\; r^2\; =\; s^3\; =\; t^3\; =\; rst\; angle$or equivalently,:$langle\; s,t\; mid\; (st)^2\; =\; s^3\; =\; t^3\; angle.$Generators with these relations are given by:$s\; =\; frac\{1\}\{2\}(1+i+j+k)\; qquad\; t\; =\; frac\{1\}\{2\}(1+i+j-k).$

**ubgroups**The

quaternion group consisting of the 8Lipschitz unit s forms anormal subgroup of 2"T" of index 3. This group and the center {±1} are the only nontrivial normal subgroups.All other subgroups of 2"T" are

cyclic group s generated by the various elements, with orders 3, 4, and 6.**ee also***

binary polyhedral group

*binary cyclic group

*binary dihedral group

*binary octahedral 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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