Groups of order 88

This article gives information about, and links to more details on, groups of order 88
See pages on algebraic structures of order 88 | See pages on groups of a particular order

Statistics at a glance

The number 88 has the prime factorization:

$\!88=2^{3}\cdot 11^{1}=8\cdot 11$

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	12
Number of abelian groups (i.e., finite abelian groups) up to isomorphism	3	(number of abelian groups of order $2^{3}$) times (number of abelian groups of order $11^{1}$) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 1) = $3\times 1\times 1=3$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism	5	(number of groups of order 8) times (number of groups of order 11) = $5\times 11=5$.
Number of solvable groups (i.e., finite solvable groups) up to isomorphism	12	All groups of this order are solvable.
Number of simple groups up to isomorphism	0	Follows from all groups of this order being solvable.

Minimal order attaining number

$88$ is the smallest number such that there are precisely $12$ groups of that order up to isomorphism. That is, the value of the minimal order attaining function at $12$ is $88$.

GAP implementation

The order 88 is part of GAP's SmallGroup library. Hence, any group of order 88 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 88 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function: