I do not have access to Magma, so I do not now how exactly the Small Groups Library in Magma differs from the one in GAP, and whether missing orders that you observe using the online calculator are limitations of the online calculator or are also missing in a proper Magma installation.
As a test case, I've tried $n=3 \cdot 17^2 \cdot 25= 21675$ and Magma online calculator returns
Runtime error: The groups of order 21675 are not contained in the small groups database. In GAP,
NrSmallGroups(n) unsurprisingly returns 9, matching Small Groups Library page which says that it contains groups of cubefree order at most 50 000 (395 703 groups).
There is no GAP online calculator, at least at the moment, but you may try to access GAP via SageMathCloud. I've just tried to open terminal there and called
gap. This started GAP 4.7.8 with some selection of packages (which is however smaller than in the GAP installation from sources downloaded from the GAP website and fully compiled, i.e. with packages that require compilation, so your mileage may vary. Of course you may install extra packages or even install your own full copy of GAP there, but then you would have to maintain it yourself). Also, SageMathCloud suggests to consider subscription since "projects running on free servers compete for resources with a large number of other free users".
Hope this helps to create a setup which works well for you, perhaps using a combination of working with local GAP installation when you can, and with SageMathCloud when you are travelling with internet connection.
Update: GAP package SCSCP implements remote procedure call protocol, also called SCSCP. Using this protocol, SCSCP-compliant computer algebra systems (and other compliant software) can communicate with each other.
NrSmallGroups has been just added to the list of procedures provided by the demo server, so now (important! if IO package is compiled and SCSCP package is loaded) one could read into GAP the following auxiliary function
NrSmallGroupsRemote := n -> EvaluateBySCSCP( "NrSmallGroups", [n],
and then use it as follows:
[ 49487365422, 408641062, 9 ]
If you will ask for the order where no information is available, you will get an error message, for example:
Error, chrystal.mcs.st-andrews.ac.uk.0.0.0.0:26133 reports :
the library of groups of size 2048 is not available
cd := "scscp1",
name := "error_system_specific" ) called from
For cube-free groups of order greater than 50000, the GAP SCSCP server now provides two functions from the CubeFree package:
NumberCFSolvableGroups. You can call them remotely as follows (after
rec( attributes := [ [ "call_id", "chrystal.mcs.st-andrews.ac.uk.0.0.0.0:26133:12088:c7Ikdmtp" ] ],
object := 1 )
rec( attributes := [ [ "call_id", "chrystal.mcs.st-andrews.ac.uk.0.0.0.0:26133:12088:n3sJKTbZ" ] ],
object := 2 )
The number of the groups is the
object component of the returned record, the rest is technical information that may be ignored.
So in theory one could have a minimalistic GAP installation with stripped off data libraries and only 4 packages: GAPDoc, IO, OpenMath and SCSCP, which will do the job.
However, SCSCP is a protocol implemented in several systems and languages - see the list of compatible tools here, so it's completely plausible that instead of GAP you may have e.g. a short Python script that will perform only a single purpose - contact GAP SCSCP server sending one integer (the order of a group) and getting back another integer (the number of groups of these order) or an error message.