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We calculate the character table of a sharply 5-transitive subgroup of Alt(12), and of a sharply 4-transitive subgroup of Alt(11). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.

Let G be a , nite group which is not cyclic of prime power order. The join graph ∆, (G) of G is a graph whose vertex set is the set of all proper subgroups of G, which are not contained in the Frattini subgroup G and two distinct vertices H and K are adjacent if and only if G = ⟨,H,K ⟩, . Among other results, we show that if G is a , nite cyclic group and H is a , nite group such that ∆, (G) ,= ∆, (H), then H is cyclic. Also we prove that ∆, (G) ,= ∆, (A5) if and only if G ,= A5.

In this paper we characterize the finite groups with an irredundant covering containing some p-Sylow subgroups. In particular we analyze the symmetric and Alternating groups, finding their p-elements having a p-subgroup as centralizer.

We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the Alternating group $A_7$. It is shown that a self-dual symmetric 2-$(35,18,9)$ design and an optimal even binary $[21,14,4]$ LCD code are found such that they are invariant under the full automorphism groups $S_8$ and $S_7$, respectively. Moreover, designs with parameters 1-$(21,l,k_{1,l})$ and 1-$(35,l,k_{2,l})$ are obtained, where $\omega$ is a codeword, $l=wt(\omega)$, $k_{1,l}=l|\omega^{S_7}|/21$ and $k_{2,l}=l|\omega^{S_7}|/35$. It is seen that there exist a 2-$(21,5,12)$ design with the full automorphism group $S_7$ among these 1-designs.