Groups of order 70

This article gives information about, and links to more details on, groups of order 70
See pages on algebraic structures of order 70 | See pages on groups of a particular order

Statistics at a glance

The number 70 has prime factors 2, 5, and 7. The prime factorization is:

$70=2^{1}\cdot 5^{1}\cdot 7^{1}$

One such way to classify groups of order 70 is therefore by the classification of groups of order 2pq.

Square-free implies solvability-forcing, so all groups of order 70 are finite solvable groups. Moreover, every Sylow subgroup is cyclic implies metacyclic, so all groups of order 70 are in fact metacyclic groups.

The list

There are 4 groups of order 70:

Group	Second part of GAP ID	Abelian	Direct Product
direct product of dihedral group:D10 and cyclic group:Z7	1	no	yes
direct product of dihedral group:D14 and cyclic group:Z5	2	no	yes
dihedral group:D70	3	no	no
cyclic group:Z70	4	yes	yes