That IP address is not of any of the IP blocks reserved for internal/private networking.
Suggest you install NMap on the laptop.
http://nmap.org/book/inst-windows.html
And then use this command to see if you can access the server when your laptop is connected to a network outside the private network:
nmap -P0 -p 80 134.26.65.163
that's a -PZero not an "O"
That scans the ip address to see if port 80 is open and available. If the scan returns 'Host is up' but you see 'filtered' it means there is restriction on port 80. Your browser will not be able to see anything served out from the server ... it will time out.
Two possbilities ... CentOS server itself has a firewall that could restrict access to port 80. But since you evidently have worked on the Moodle while inside the private network it's doubtful that firewall is restricting. If that is true, then it's the boundary/network firewall that is not granting access to the Moodle server.
With the laptop plugged in/connected to the private network, using nmap again, use the following command replacing appropriate values:
nmap -P0 -p 80 [internalIPaddressofMoodleServer]
Besides that, there still needs to be a DNS entry for a Moodle 2 to work. There needs to be a DNS entry in both external DNS and in internal DNS. Currently, there is no DNS entry in external DNS for the IP address you've given.
What's in your config.php file (following is a working example):
$CFG->wwwroot = 'http://sos.tcea.org/moodle24';
And, check the httpd.conf file for apache found in /etc/httpd/conf/ for the following (the # symbols are comment lines):
# If your host doesn't have a registered DNS name, enter its IP address here.
# You will have to access it by its address anyway, and this will make
# redirections work in a sensible way.
#
#ServerName www.example.com:80
Does the line for ServerName have an IP address? or is it commented out like above?
It should be commented out.
'spirit of sharing', Ken