Groups of order 120

This article gives information about, and links to more details on, groups of order 120
See pages on algebraic structures of order 120 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 120. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 120
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 120
subgroup structure	subgroup structure of groups of order 120
linear representation theory	linear representation theory of groups of order 120 projective representation theory of groups of order 120 modular representation theory of groups of order 120
endomorphism structure, automorphism structure	endomorphism structure of groups of order 120
group cohomology	group cohomology of groups of order 120

Statistics at a glance

The number 120 has prime factorization $120=2^3\cdot 3\cdot 5$. There are solvable as well as non-solvable groups of this order. All the non-solvable groups have alternating group:A5 and cyclic group:Z2 as the composition factors in their composition series.

Quantity	Value	List/comment
Total number of groups	47
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 $3^1$) times (number of abelian groups of order $5^1$) = $3\times 1\times 1=3$. 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 3) times (number of groups of order 5) = $5\times 1\times 1=5$. 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 (i.e., finite solvable groups) up to isomorphism	44	See note on non-solvable groups
Number of non-solvable groups up to isomorphism	3	The three non-solvable groups are special linear group:SL(2,5), direct product of A5 and Z2, and symmetric group:S5
Number of simple groups up to isomorphism	0
Number of almost simple groups up to isomorphism	1	symmetric group:S5 (isomorphic to $PGL(2,5)$)
Number of quasisimple groups	1	special linear group:SL(2,5) (also the binary icosahedral group)
Number of almost quasisimple groups up to isomorphism	2	symmetric group:S5 and special linear group:SL(2,5)
Number of perfect groups up to isomorphism	1	special linear group:SL(2,5)

Classification of non-solvable groups

The classification proceeds in steps, which are presented in sequence for clarity:

Step no.	What we are trying to find	What we conclude	Explanation
1	All the possibilities for simple non-abelian group of order dividing 120	the only simple non-abelian group is alternating group:A5, and it has order 60	This follows from A5 is the simple non-abelian group of smallest order and the fact that there is no simple non-abelian group of order 120.
2	All the possibilities for the (unordered) collection of composition factors of a non-solvable group of order 120	one occurrence of alternating group:A5 and one occurrence of cyclic group:Z2	At least one of the composition factors must be simple non-abelian for the group to be non-solvable. So one slot goes to alternating group:A5. This takes up 60 of the 120, leaving $120/60=2$, which must be taken up by cyclic group:Z2.
3	All the possibilities for the composition series of a group of order 120	normal subgroup isomorphic to cyclic group:Z2, quotient group isomorphic to alternating group:A5; OR normal subgroup isomorphic to alternating group:A5, quotient group isomorphic to cyclic group:Z2	Direct from Step (2)
4.1	All the possibilities for a group of order 120 with a normal subgroup isomorphic to alternating group:A5 and quotient group isomorphic to cyclic group:Z2	direct product of A5 and Z2 (trivial case), symmetric group:S5 (almost simple group case)	[SHOW MORE] In this case, because quotient group maps to outer automorphism group of normal subgroup, we get a mapping from ${\mathbb{Z}}_2$ to $\mathrm{O}\mathrm{u}\mathrm{t}(A_5)$, which is $S_5/A_5\cong {\mathbb{Z}}_2$. In other words, the possibilities are classified by $\mathrm{H}\mathrm{o}\mathrm{m}({\mathbb{Z}}_2,{\mathbb{Z}}_2)$. If the map is trivial, we get direct product of A5 and Z2 and if the map is nontrivial, we get symmetric group:S5. (This is the almost simple case: where the outer action map is faithful).
4.2	All the possibilities for a group of order 120 with a normal subgroup isomorphic to cyclic group:Z2 and quotient group isomorphic to alternating group:A5	direct product of A5 and Z2, special linear group:SL(2,5) (quasisimple group case)	[SHOW MORE] We see that ${\mathbb{Z}}_2$ must be in the center, because the induced action of the quotient is trivial. Thus, the possibilities are classified by $H^2(A_5;{\mathbb{Z}}_2)$ which is cyclic group:Z2 (see second cohomology group for trivial group action of A5 on Z2). The identity element corresponds to direct product of A5 and Z2 and the non-identity element corresponds to special linear group:SL(2,5). Another way of thinking of this is that it corresponds to homomorphisms from the Schur multiplier of $A_5$ (which is ${\mathbb{Z}}_2$) to ${\mathbb{Z}}_2$. The trivial homomorphism gives the direct product, whereas the identity map gives $SL(2,5)$, which is the universal central extension (i.e., the Schur covering group). The latter case gives a quasisimple group as it arises from a surjective homomorphism.
5	Overall conclusion	direct product of A5 and Z2 (occurring in both cases), symmetric group:S5, and special linear group:SL(2,5)	Combine Steps (3), (4.1), and (4.2)

GAP implementation

The order 120 is part of GAP's SmallGroup library. Hence, any group of order 120 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 120 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(120);

  There are 47 groups of order 120.
  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 has Frattini factor [ 60, 5 ].
     6 has Frattini factor [ 60, 6 ].
     7 has Frattini factor [ 60, 7 ].
     8 - 14 have Frattini factor [ 60, 8 ].
     15 has Frattini factor [ 60, 9 ].
     16 - 20 have Frattini factor [ 60, 10 ].
     21 - 25 have Frattini factor [ 60, 11 ].
     26 - 30 have Frattini factor [ 60, 12 ].
     31 - 33 have Frattini factor [ 60, 13 ].
     34 - 47 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.