Groups of order 68

This article gives information about, and links to more details on, groups of order 68
See pages on algebraic structures of order 68 | See pages on groups of a particular order

Statistics at a glance

The number 68 has prime factors 2 and 17. The prime factorization is as follows:

$\!20=2^{2}\cdot 17^{1}=4\cdot 17$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity	Value	Explanation
Total number of groups up to isomorphism	5	See classification of groups of order four times a prime congruent to 1 modulo four.
Number of abelian groups up to isomorphism	2	(number of abelian groups of order $2^{2}$ ) $\times$ (number of abelian groups of order $17^{1}$ ) = (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 1) = $2\times 1=2$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism	2	(number of groups of order 4) $\times$ (number of groups of order 17) = $2\times 1=2$ . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. See also nilpotent of cube-free order implies abelian.
Number of solvable groups up to isomorphism	5	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups up to isomorphism	0	All groups of this order are solvable

The list

Group	Second part of GAP ID (GAP ID is (68,second part))	abelian?
dicyclic group:Dic68	1	No
cyclic group:Z68	2	Yes
semidirect product of Z17 and Z4	3	No
dihedral group:D68	4	No
direct product of Z34 and Z2	5	Yes