Groups of order 80

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This article gives information about, and links to more details on, groups of order 80
See pages on algebraic structures of order 80| See pages on groups of a particular order

Statistics at a glance

The number 80 has prime factorization 80 = 2^4 \cdot 5. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 80 are solvable groups (specifically, finite solvable groups).

Quantity Value Explanation
number of groups up to isomorphism 52
number of abelian groups up to isomorphism 5 (number of abelian groups of order 2^4) times (number of abelian groups of order 5^1) = (number of unordered integer partitions of 4) times (number of unordered integer partitions of 1) = 5 \times 1 = 5. See also classification of finite abelian groups
number of nilpotent groups up to isomorphism 14 (number of groups of order 16) times (number of groups of order 5) = 14 \times 1 = 14. 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.
number of solvable groups up to isomorphism 52 There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 80 are solvable groups (specifically, finite solvable groups).
number of simple groups up to isomorphism 0 All groups of this order are solvable, so there cannot be any simple groups.

GAP implementation

The order 80 is part of GAP's SmallGroup library. Hence, all groups of order 80 can be constructed using the SmallGroup function and have group IDs. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

gap> SmallGroupsInformation(80);

  There are 52 groups of order 80.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 10, 1 ].
     2 has Frattini factor [ 10, 2 ].
     3 has Frattini factor [ 20, 3 ].
     4 - 19 have Frattini factor [ 20, 4 ].
     20 - 27 have Frattini factor [ 20, 5 ].
     28 - 34 have Frattini factor [ 40, 12 ].
     35 - 44 have Frattini factor [ 40, 13 ].
     45 - 48 have Frattini factor [ 40, 14 ].
     49 - 52 have trivial Frattini subgroup.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
     LGLength, FrattinifactorSize and FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.