I'm trying to understand a proof in "Groups with a maximal irredundant 6-cover" by A. Abdollahi et al, page 3231, Lemma 4.1 itens (2) and (3).
Here's my problem:
Lemma 4.1
(2) The following are $\mathfrak{G}_6$-groups:
(d) $G \cong (C_5 \times C_5) \rtimes C_4$ with $Z(G) = 1$. In this case, the intersection of a maximal irredundant cover which is core-free has size 2.
(3) A subdirect product G of three symmetric groups $Sym_3$ is a $\mathfrak{G}_6$-groups if and only if G is isomorphic to one the following groups: $(C_3 \times C_3) \rtimes C_2$, $S_3 \times S_3$, $(C_3)^3 \rtimes C_2$, where the size of the intersection of an arbitrary maximal irredundant 6-cover with core-free intersection is 1, 1, and 2, respectively.
Remark:
$(*)$ G is $\mathfrak{G}_6$-groups mean $G = M_1 \cup ...\cup M_6$ is a maximal irredundant 6-cover with core-free intersection.
$(**)$ In outline proof is used the following function written with GAP (2002) program
f:= function(G) local
S, M, n, C, i, T, Q, R;
n:=Size(G); M:=MaximalSubgroups(G);
C:=Combinations(M,6);
S:=[]; for i in [1..Size(C)] do if
Size(Union(C[i]))=n then Add(S, C[i]); fi;
od; T := []; for i in
[1..Size(S)] do if Size(Core(G, Intersection(S[i])))=1 then
Add(T,S[i]); fi;
od; R:=[]; for i in [1..Size(T)] do
Q:=Combinations(T[i],5); if (n in
List(Q, i->Size(Union(i))))=false then Add(R,T[i]); fi; od;
return R; end;
The input of the function is a group G and the outputs are all the irredundant 6-covers with core-free intersection for G, and if there is no such a cover for G, the output is the empty list.
How to apply to items (2) and (3) in the function f written in the GAP to give a demonstration?