## Program Extensions

### Usage

Extensions ['file1'] 'file2' ['file3'] [-n] [-i] [-t=n] [-v] [-C] [-H] [-F] [-S]

file1: matrix_TYP containing a presentation of the group (cf. Presentation)

file2: bravais_TYP containing the group (and its normalizer).

file3: (OPTIONAL) containing cocycles for identification.

### Description

Calculates representatives of the isomorphism classes of extension of
the group in file2 with presentation in file1 with the natural lattice.
Output lists representative vectorsystems. (Output can be used as input
for Extract). Also the isomorphism type of the extension group is
given.
### Options

-n: only calculates the number of extensions, without computing
representatives. This option is much faster for big cohomology
groups. WARNING: Try this option first for 2-groups in dimension
greater than 4.
-v: verbose mode. Give some echoing to stderr to
indicate a little what the program is doing.
-i: Identify the cozycles given in file3, ie. give
the described space groups a name. CAUTION: The
name will depend on the generating set of the group
in file2 and the presentation in file1.
Can be used to test isomorphism of space groups with equal
point groups. The name is 0 iff the extension splits.
-t=n: Has an effect only if given with -i. Outputs the
isomophism needed to transform the space group
By default, only the linear part is calculated. To
get a full transformation matrix, use -t=2.
-C: Ignore the operation of the normalizer, just work
on the level of extensions.
-H: echo the isomorphism type of the cohomology group
H^1(G,Q^n/Z^n) to stderr.
-F: Only construct those extensions which gie rise to
torsion free space groups. Does not work in conjunction
with -n.
-S: write corresponding space groups in files

### Remarks

THE PROGRAM RELIES HEAVILY ON THE FOLLOWING TWO FACTS:

- THE PRESENTATION GIVEN IN file1 IS INDEED A PRESENTATION. CHECKED IS ONLY THAT THE GIVEN
GENERATORS FULFILL THE RELATIONS.
- THE NORMALIZER IN GL
_{n}(Z) FOR THE GROUP IN file2 IS CORRECT. THE PROGRAM MIGHT NOT
FINISH OR GIVE WRONG ANSWERS OTHERWISE.

'file1' can be left out, but the program is faster, if the presentation is given.

See also for Extract, Presentation,
Same_generators and Zass_main.

Extensions is a synonym for Vector_systems.

### Examples

- Find all space groups with a given point group and decide for
which superlattices each extension splits.
- Find all Z-classes, affine classes and torsion free space
groups in a given Q-class.