Dear Nicola, dear GAP-forum

I think the following should compute your desired id's:

# create the group as permutation group
gap> f := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> g:=f/[f.1^3, f.2^7, f.3^16, (f.1*f.2)^2, (f.2*f.3)^2, (f.3*f.1)^2,
> (f.1*f.2*f.3)^2];
<fp group on the generators [ f1, f2, f3 ]>
gap> G := Image( IsomorphismPermGroup(g) );
<permutation group with 3 generators>
gap> Size(G);
21504

# get subgroups of order 192
gap> f := function( G ) return IsInt( 192/Size(G) ); end;
function( G ) ... end
gap> lat := LatticeByCyclicExtension( G, f, true );
<subgroup lattice of <permutation group of size 21504 with 3 generators>,
740 classes, 125227 subgroups, restricted under further condition l!.func>
gap> cl := lat!.conjugacyClassesSubgroups;;
gap> Length(cl);
740
gap> rep := List( cl, Representative );;
gap> sub := Filtered( rep, x -> Size(x) = 192 );;

# get their id's
gap> Length(sub);
19
gap> List( sub, IdGroup );
[ [ 192, 1540 ], [ 192, 1000 ], [ 192, 191 ], [ 192, 1538 ], [ 192, 1538 ],
  [ 192, 1538 ], [ 192, 1495 ], [ 192, 1538 ], [ 192, 1538 ], [ 192, 955 ],
  [ 192, 955 ], [ 192, 181 ], [ 192, 198 ], [ 192, 181 ], [ 192, 198 ],
  [ 192, 181 ], [ 192, 181 ], [ 192, 1472 ], [ 192, 1002 ] ]


Best wishes,
            Bettina