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This paper is dealing with a split extension group of the form 26: (3,A5),which is the largest maximal subgroup of the Symplectic group Sp(4,4): We refer to this extension by G: We , rstly determine the conjugacy classes of G using the coset analysis technique. The structures of inertia factor groups were determined. We then compute the Fischer matrices of G and apply the Cli, ord-Fischer theory to calculate the ordinary character table of this group. The Fischer matrices of G are all integer valued, with sizes ranging from 1 to 4. The full character table of G is 26, 26 complex valued matrix and is given at the end of this paper.

A finite group G is called (l,m,n)-generated, if it is a quotient group of the triangle group T(l,m,n) = ⟨,x,y,zjxl = ym = zn = xyz = 1 ⟩, : In [29], Moori posed the ques-tion of , nding all the (p,q,r) triples, where p,q and r are prime numbers, such that a non-abelian finite simple group G is a (p,q,r)-generated. In this paper we establish all the (p,q,r)-generations of the symplectic group Sp(6,2): GAP [20] and the Atlas of finite group representations [33] are used in our computations.

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on the leaves of the distribution. Hence, the induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with an invariant metric and an admissible almost complex structure, we show that the bracket annihilates on the kernel of the anchor map.

In this paper I will determine all of the JS - maximal soluble subgroups of the groups GL (18; pk), GL (20; pk), and GL (21; pk). It turns out that the number of types of the JS - maximal soluble subgroups in the groups GL (18; pk), GL (20; pk), and GL (21; pk) are 21, 20 and 10 respectively. Furthermore, we find the structure of these subgroups.

We study the notion of harmonicity in the sense of symplectic geometry, and investigate the geometric properties of harmonic Thom forms and distributional Thom currents, dual to different types of submanifolds. We show that the harmonic Thom form associated to a symplectic submanifold is nowhere vanishing. We also construct symplectic smoothing operators which preserve the harmonicity of distributional currents and using these operators, construct harmonic Thom forms for co-isotropic submanifolds, which unlike the harmonic forms associated with symplectic submanifolds, are supported in an arbitrary tubular neighborhood of the manifold.

We study the automorphic theta representation Q2n(r) on the r-fold cover of the symplectic group Sp2n. This representation is ob-tained from the residues of Eisenstein series on this group. Ifr is odd, n£ r<2n, then under a natural hypothesis on the theta representations, we show that Q2n(r) may be used to construct a globally generic represen-tation s(2r) 2n-r+1 on the 2r -fold cover of Sp 2n-r+1. Moreover, when r=n the Whittaker functions of this representation attached to factorizable data are factorizable, and the unramied local factors may be computed in terms ofn -th order Gauss sums. If n=3 we prove these results, which in that case pertain to the six-fold cover of Sp 4, unconditionally. We expect that in fact the representation constructed here, s(2r) 2n-r+1, is precisely Q(2r) 2n-r+1; that is, we conjecture relations between theta repre-sentations on different covering groups.

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In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementary particles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their representations on these algebras can be described easily.