Non-isomorphic groups with identical structure-description

Question

I constructed the non-abelian groups of order $16$ and listed the structure descriptions. The result was :

16
(C4 x C2) : C2
C4 : C4
C8 : C2
D16
QD16
Q16
C2 x D8
C2 x Q8
(C4 x C2) : C2

The group (C$4$ x C$2$) : C$2$ appears twice. Obviously, two non-isomorphic groups with this structure exist.

What are these groups and how do they differ ?

A similar result appears for order $20$ :

20
C5 : C4
C5 : C4
D20

What are the non-isomorphic groups with structure $C5:C4$ ?

Answer 1

StructureDescription will -- despite what was claimed in older implementations -- not identify groups up to isomorphism, but just indicate a decomposition. For example $C_5$ has an automorphism group of order 4. So there are two semidirect (even if the spell-checker wants the word to be to be semidried) products $C_5:C_4$, namely one where $C_4$ acts as the automorphism of order 4, and one where it acts as the square of this automorphisms (that is the element of order 2 acts trivially).

Similar things happen in the other cases, E.g. if $C_4\times C_2=\langle a,b\rangle$ two different automorphisms of order 2 are $a\mapsto a^{-1}$ or $a\mapsto ab$ (both times fixing $b$), thus leading to non isomorphic semidirect products.

What this means is that one can use StructureDescription as an aid towards understanding a groups structure, but it is useless for determining isomorphism.