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Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1.
Cayley table as general (and special) linear group GL(2, 2) In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1] This page illustrates many group concepts using this group as example.
S 6 has exactly one (class) of outer automorphisms: Out(S 6) = C 2. To see this, observe that there are only two conjugacy classes of S 6 of size 15: the transpositions and those of class 2 3. Each element of Aut(S 6) either preserves each of these conjugacy classes, or exchanges them. Any representative of the outer automorphism constructed ...
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The dihedral group D 2 is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D 2 can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis. The four elements of D 2 (x-axis is vertical here) D 2 is isomorphic to the Klein ...
The conjugacy classes of the full octahedral group, O h ≅ S 4 × C 2, are: inversion; 6 × rotoreflection by 90° 8 × rotoreflection by 60° 3 × reflection in a plane perpendicular to a 4-fold axis; 6 × reflection in a plane perpendicular to a 2-fold axis; The conjugacy classes of full icosahedral symmetry, I h ≅ A 5 × C 2, include also ...
McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A 8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A 8 .
The point stabilizer is sometimes denoted by M 10, and is a non-split extension of the form A 6.2 (an extension of the group of order 2 by the alternating group A 6). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.