A book of John Conway suggests there are 221 groups of order $|G| = 400$. How do I go about finding these. Commutative groups with $ab = ba$ can be listed very easily:
- $\mathbb{Z}/400\mathbb{Z}$
- $\mathbb{Z}/200\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
- $\mathbb{Z}/25\mathbb{Z} \oplus \mathbb{Z}/16\mathbb{Z}$
- ...
There are 207 super-solvable groups of order $|G| = 400$ How do we list some of them?
$$ 1 \leq H_0 \leq H_1 \leq H_2 \leq \dots \leq H_n = G $$
here $H_i \vartriangleleft G$ and $H_{i+1}/H_i$ is cyclic. This could be a great way to explain the difference between nilpotent and solvable groups. Some discussion here however I will put in the tag representation-theory which includes (for example) matrix representation or permutation representations.
There are 28 nilpotent groups of order $|G| = 400$.
I haven't used GAP the question would be how does the computer program find such objects? Here's some of what it found:
- $G = (C_5 \ltimes Q_8 ) \times D_{10}$
- $G = (C_5 \ltimes C_5) \ltimes (C_4 \times C_4) $
- $G = C_2 \times ((C_5 \times C_5) \ltimes C_8)$
These names or descriptions leave it upon us to say what these symmetries actually look like.
- Example, $D_{10}$ is the dihedral group or the symmetry group of a 10-gon
- Also $D_{10} = C_{10} \ltimes C_2 $ , see also [1] .
- $C_n \simeq \mathbb{Z}/n\mathbb{Z}$ is the cyclic group .
Using the SmallGrp package for GAP and the function
StructureDescriptionyou can get some insight into the structure of the groups of order 400. See the documentation for how to interpret these strings, e.g. a colon:denotes a semidirect product.