SINIF CEBİRİ YAKLAŞIMI ALTINDA 𝑫𝟐𝒅 ve 𝑪𝟑𝒊 NOKTA GRUPLARI

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Layer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Server

The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouin-zone databases for the layer groups. The crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouin-zone database (LKVEC) offers k-vector tables and Brillouin-zone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the so-called reciprocal-space-group approach and this classification scheme is compared with that of Litvin & Wike [(1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press]. The specification of independent parameter ranges of k vectors in the representation domains of the Brillouin zones provides a solution to the problems of uniqueness and completeness of layer-group representations. The Brillouin-zone figures and k-vector tables are described in detail and illustrated by several examples.

A revision of the genus Psammogorgia Verrill, 1868 (Cnidaria, Anthozoa, Octocorallia) in the tropical eastern Pacific Ocean

The species of the genus Psammogorgia Verrill, 1868 from the shallow waters of the tropical eastern Pacific were mainly described from 1846 to 1870. Very few contributions were published subsequently. Recently, the genus was revisited with the addition of two new species. However, a comprehensive generic study is still missing for the eastern Pacific. Psammogorgia is characterised by having axes cores without mineralisation, mainly coarse irregular spindles and thorny, leafy or tuberculate clubs coenenchymal sclerites and the anthocodial armature with distinct collaret and points arrangements. Herein a taxonomic revision of the genus is presented based on type material which was morphologically analysed and illustrated using optical and scanning electron microscopy. Comparative character tables are provided for comparison among species in the genus, along with a taxonomic key. Moreover, the taxonomic status of each species was analysed. The genus Psammogorgia comprises six valid species and two varieties, and three lectotypes and a new combination are proposed to establish the taxonomic status of these species.

The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.