Groups of order 90

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This article gives information about, and links to more details on, groups of order 90
See pages on algebraic structures of order 90| See pages on groups of a particular order

Statistics at a glance

The number 90 has prime factors 2, 3, and 5. The prime factorization is as follows:

\! 90 = 2 \cdot 3^2 \cdot 5 = 2 \cdot 9 \cdot 5

All groups of this order are finite solvable groups.

GAP implementation

The order 90 is part of GAP's SmallGroup library. Hence, any group of order 90 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 90 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:

gap> SmallGroupsInformation(90);

  There are 10 groups of order 90.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 30, 1 ].
     2 has Frattini factor [ 30, 2 ].
     3 has Frattini factor [ 30, 3 ].
     4 has Frattini factor [ 30, 4 ].
     5 - 10 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.