You are confusing arrays and hashes, Perl is not PHP ;-)

You can only use integers to index arrays, you can only use strings to index hashes.

You can use (paraphrasing your code)

use strict;
use Data::Dumper;
my %data;
my $name = "group 0";
for my $i (0 .. 2) {
  for my $j (0 .. 3) {
    $data{$name}[$i][$j] = "group 0 - index $i - $j";
  }
}
print Dumper(\%data);

# output follows
__END__


$VAR1 = {
          'group 0' => [
                         [
                           'group 0 - index 0 - 0',
                           'group 0 - index 0 - 1',
                           'group 0 - index 0 - 2',
                           'group 0 - index 0 - 3'
                         ],
                         [
                           'group 0 - index 1 - 0',
                           'group 0 - index 1 - 1',
                           'group 0 - index 1 - 2',
                           'group 0 - index 1 - 3'
                         ],
                         [
                           'group 0 - index 2 - 0',
                           'group 0 - index 2 - 1',
                           'group 0 - index 2 - 2',
                           'group 0 - index 2 - 3'
                         ]
                       ]
        };
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See perldata for much more information.