Groups of order 100

This article gives information about, and links to more details on, groups of order 100
See pages on algebraic structures of order 100 | See pages on groups of a particular order

Statistics at a glance

The number 100 has 2 and 5 as its only prime factors. The prime factorization is as follows:

$\!100=2^{2}\cdot 5^{2}=4\cdot 25$

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	16
Number of abelian groups up to isomorphism	4	(number of abelian groups of order $2^{2}$)$\times$(number of abelian groups of order $5^{2}$) = (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 2) = $2\times 2=4$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism	4	(number of groups of order 4) $\times$ (number of groups of order 25) = $2\times 2=4$. 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 supersolvable groups up to isomorphism	16
Number of solvable groups up to isomorphism	16

The list

Group	GAP ID (second part)	Abelian?
dicyclic group:Dic100	1	No
cyclic group:Z100	2	Yes
semidirect product of Z25 and Z4	3	No
dihedral group:D100	4	No
direct product of Z2 and Z50	5	Yes
direct product of Z5 and Dic20	6	No
SmallGroup(100,7)	7	No
direct product of Z5 and Z20	8	Yes
direct product of Z5 and GA(1,5)	9	No
SmallGroup(100,10)	10	No
SmallGroup(100,11)	11	No
SmallGroup(100,12)	12	No
direct product of D10 and D10	13	No
direct product of Z10 and D10	14	No
SmallGroup(100,15)	15	No
elementary abelian group:E100	16	Yes

Minimal order attaining number

$100$ is the smallest number such that there are precisely $16$ groups of that order up to isomorphism. That is, the value of the minimal order attaining function at $16$ is $100$.

GAP implementation

The order 100 is part of GAP's SmallGroup library. Hence, any group of order 100 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 100 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(100);

  There are 16 groups of order 100.
  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 has Frattini factor [ 20, 4 ].
     5 has Frattini factor [ 20, 5 ].
     6 has Frattini factor [ 50, 3 ].
     7 has Frattini factor [ 50, 4 ].
     8 has Frattini factor [ 50, 5 ].
     9 - 16 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.