# Groups of order 120

See pages on algebraic structures of order 120| See pages on groups of a particular order

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

### Factorization and useful forms

The number 120 has prime factors 2,3,and 5, and factorization: $120 = 2^3 \cdot 3^1 \cdot 5^1 = 8 \cdot 3 \cdot 5$.

Other expressions for this number are: $120 = 5! = 5^3 - 5 = 2(4^3 - 4) = \frac{4}{\frac{1}{2} + \frac{1}{3} + \frac{1}{5} - 1}$

### Group counts

Quantity Value Explanation
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. Note that all of them have alternating group:A5 and cyclic group:Z2 as composition factors. For more, see the section #Classification of non-solvable groups.
Number of possibilities for the multiset of composition factors (i.e., number of equivalence classes under composition factor-equivalence) 2 The equivalence class for finite solvable groups has cyclic group:Z2 (3 times), cyclic group:Z3 (1 time), cyclic group:Z5 (1 time)
There is a unique equivalence class for non-solvable groups, with alternating group:A5 (order 60, 1 time) and cyclic group:Z2 (1 time)
See order of group is product of orders of composition factors and classification of possible multisets of composition factors for groups of a given order.
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 up to isomorphism 1 special linear group:SL(2,5) (also called the binary icosahedral group and also isomorphic to the double cover of alternating group $2 \cdot A_5$)
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]
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]
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)

## Relation with other orders

### Divisors of the order

More in-depth information can be found under subgroup structure of groups of order 120.

Divisor Quotient value Number of groups of the order Information on groups of the order Relationship (subgroup perspective) Relationship (quotient value)
2 60 1 cyclic group:Z2
3 40 1 cyclic group:Z3
4 30 2 groups of order 4
5 24 1 cyclic group:Z5
6 20 2 groups of order 6
8 15 5 groups of order 8
10 12 2 groups of order 10
12 10 5 groups of order 12
15 8 1 cyclic group:Z15
20 6 5 groups of order 20
24 5 15 groups of order 24
30 4 4 groups of order 30
40 3 14 groups of order 40
60 2 13 groups of order 60

### Multiples of the order

Related in-depth information can be found under supergroups of groups of order 120.

Multiplier (other factor) Multiple Number of groups Information on groups of the order Relationship (subgroup perspective) Relationship (quotient perspective)
2 240 208 groups of order 240
3 360 162 groups of order 360
4 480 1213 groups of order 480
5 600 205 groups of order 600
6 720 840 groups of order 720
7 840 186 groups of order 840
8 960 11394 groups of order 960
9 1080 583 groups of order 1080
10 1200 1040 groups of order 1200

## 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.