M24 is the largest Mathieu sporadic simple group of order 244823040=210.33.5.7.11.23 contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of M24 with respect to the conjugacy classes of all its nonidentity elements.

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We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.

The commuting conjugacy class graph of a non-abelian group G, denoted by CCC(G), is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of G and two distinct vertices xG and yG are adjacent if there exists some elements x ′,2 xG and y ′,2 yG such that x ′,y ′,= y ′,x ′, . In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups D2n, Q4m, U(n, m),V8n and SD8n. Our computation shows that CCC(G) is super integral for these groups. We compare various energies and as a consequence it is observed that CCC(G) satisfy E-LE Conjecture of Gutman et al. We also provide negative answer to a question posed by Dutta et al. comparing Laplacian and Signless Laplacian energy. Finally, we conclude this paper by characterizing the above mentioned groups G such that CCC(G) is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.

For a non-abelian group G, its commuting conjugacy class graph CCC(G) is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of G and two distinct vertices xG and yG are adjacent if there exists some elements x ′,∈,xG and y ′,∈,yG such that x ′,y ′,= y ′,x ′, . In this paper we compute the genus of CCC(G) for six well-known classes of non-abelian two-generated groups (viz. D2n, SD8n, Q4m, V8n, U(n, m) and G(p, m, n)) and determine whether CCC(G) for these groups are planar, toroidal, doubletoroidal or triple-toroidal.

Let G be a  nite group. We say that an element g in G is a vanishing element if there exists some irreducible character  of G such that  (g) = 0. In this paper, we classify groups whose set of vanishing elements is exactly a conjugacy class.

Suppose G is a finite non-abelian group and Γ(G) is a simple graph with the non-central conjugacy classes of G as its vertex set. Two different noncentral conjugacy classes C and B are assumed to be adjacent in Γ(G) if and only if there are elements a ∈ A and b ∈ B such that ab = ba. This graph is called the commuting conjugacy class graph of G. In this paper, the structure of the commuting conjugacy class graph of a group G with this property that Z(G)/G≅D_{2n} will be determined.

In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group G(p, m, n) = ⟨x, y: x pm = y p n = [x, y] p = 1, [x, [x, y]] = [y, [x, y]] = 1⟩, where p is any prime, m ≥ 1 and n ≥ 1. We derive some consequences along with the fact that commuting conjugacy class graph of G(p, m, n) is super integral. We also compare various energies and determine whether commuting conjugacy class graph of G(p, m, n) is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.