Abstract
We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group 𝑃𝑆𝐿 2 (7).Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal.The group 2 3 .𝑃𝑆𝐿 2 (7) is a finite subgroup of the Lie Group 𝐺 2 preserving the set of octonions ±𝑒 𝑖 , (𝑖 = 1,2, … ,7) representing a 7dimensional octahedron.Its three maximal subgroups 2 3 : 7: 3, 2 3 .𝑆 4 and 4. 𝑆 4 : 2 correspond to the finite subgroups of the Lie groups 𝐺 2 , 𝑆𝑂(4) and 𝑆𝑈(3) respectively.The group 2 3 : 𝑃𝑆𝐿 2 (7) representing the split extension possesses five maximal subgroups 2 3 : 7: 3, 2 3 : 𝑆 4 , 4: 𝑆 4 : 2 and two non-conjugate Klein's group 𝑃𝑆𝐿 2 (7).The character tables of the groups and their maximal subgroups, tensor products and decompositions of their irreducible representations under the relevant maximal subgroups are identified.Possible implications in physics are discussed.
Keywords
Publication details
- DOI
- 10.24200/squjs.vol24iss1pp57-70
- Journal
- Sultan Qaboos University Journal for Science, 24(1), 57
- Publisher
- Sultan Qaboos University
- Open access
- Gold open access
- License
- CC BY 4.0
Cite this article
APA 7
Koca, M., Koç, R., & Koca, N. Ö. (2019). Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344. Sultan Qaboos University Journal for Science, 24(1), 57. https://doi.org/10.24200/squjs.vol24iss1pp57-70
MLA 9
Koca, Mehmet, et al. "Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344." Sultan Qaboos University Journal for Science, vol. 24, no. 1, 2019, pp. 57. https://doi.org/10.24200/squjs.vol24iss1pp57-70.
Chicago (author–date)
Koca, Mehmet, Ramazan Koç, and Nazife Özdeş Koca. 2019. "Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344." Sultan Qaboos University Journal for Science 24 (1): 57. https://doi.org/10.24200/squjs.vol24iss1pp57-70.
Harvard
Koca, M., Koç, R. and Koca, N. Ö. (2019) 'Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344', Sultan Qaboos University Journal for Science, 24(1), pp. 57. doi:10.24200/squjs.vol24iss1pp57-70.
Vancouver
Koca M, Koç R, Koca NÖ. Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344. Sultan Qaboos University Journal for Science. 2019;24(1):57. doi:10.24200/squjs.vol24iss1pp57-70
IEEE
M. Koca, R. Koç, and N. Ö. Koca, "Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344," Sultan Qaboos University Journal for Science, vol. 24, no. 1, pp. 57, 2019, doi: 10.24200/squjs.vol24iss1pp57-70.