# Special linear group:SL(2,5)

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## Definition

This group is defined in the following equivalent ways:

1. As the special linear group: $SL(2,5)$ is defined as the special linear group of degree two: $2 \times 2$ matrices of determinant $1$ over the field of five elements.
2. As the binary icosahedral group or binary dodecahedral group.
3. As the binary von Dyck group with parameters $(p,q,r) = (2,3,5)$.
4. As the double cover of alternating group for alternating group:A5. In other words, it is the unique stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is alternating group:A5. Viewed this way, it is denoted $2 \cdot A_5$.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 120 groups with same order As $\! SL(2,q), q = 5$ (ref: order formulas for linear groups of degree two) $\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(5-1)(5+1) = 120$
As $2 \cdot A_n, n = 5$: $2(n!/2) = n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$
As binary von Dyck group with parameters $(p,q,r) = (2,3,5)$: $\! \frac{4}{1/p + 1/q +1/r - 1} = \frac{4}{1/2 + 1/3 + 1/5 - 1} = \frac{4}{1/30} = 120$
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group As $SL(2,q), q = 5$, characteristic $p = 5$: $p(q^2 - 1)/2 = 5(5^2 - 1)/2 = 5(24)/2 = 60$
As binary von Dyck group with parameters $(p,q,r) = (2,3,5)$: $2 \operatorname{lcm}(2,3,5) = 2(30) = 60$
derived length -- not a solvable group.
nilpotency class -- not a nilpotent group.
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length The Frattini subgroup is the center, which has order two.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group --
max-length of a group 5 groups with same order and max-length of a group | groups with same max-length of a group
composition length 2 groups with same order and composition length | groups with same composition length

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of conjugacy classes 9 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As special linear group of degree two $SL(2,q), q = 5$ (odd): $q + 4 = 5 + 4 = 9$
As binary von Dyck group with parameters $(p,q,r) = (2,3,5)$: $p + q + r - 1 = 2 + 3 + 5 - 1 = 9$
As double cover of alternating group $2 \cdot A_n, n = 5$: (number of unordered integer partitions of $n$) + 3(number of partitions of $n$ into distinct odd parts) - (number of partitions of $n$ with a positive even number of even parts and with at least one repeated part) $= 7 + 3(1) - 1 = 9$

For more elaborate explanations, see element structure of special linear group:SL(2,5)#Number of conjugacy classes
number of equivalence classes under rational conjugacy 7 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy See element structure of special linear group:SL(2,5)
number of conjugacy classes of rational elements 5 groups with same order and number of conjugacy classes of rational elements | groups with same number of conjugacy classes of rational elements See element structure of special linear group:SL(2,5)
number of conjugacy classes of subgroups 12 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups See subgroup structure of special linear group:SL(2,5)
number of subgroups 76 groups with same order and number of subgroups | groups with same number of subgroups See subgroup structure of special linear group:SL(2,5)

## Group properties

### Important properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group No
perfect group Yes See special linear group is perfect
quasisimple group Yes See special linear group is quasisimple. The group is perfect, and the inner automorphism group is isomorphic to alternating group:A5, which is simple.
almost simple group No

### Other properties

Property Satisfied? Explanation
one-headed group Yes The center of order two is the unique maximal normal subgroup.
monolithic group Yes The center of order two is the unique minimal normal subgroup.
T-group Yes
Z-group No The 2-Sylow subgroup is the quaternion group, which is not cyclic.
A-group No The 2-Sylow subgroup is the quaternion group, which is not abelian.
Schur-trivial group Yes
finite group with periodic cohomology Yes the 2-Sylow subgroup is the quaternion group and the other Sylow subgroups are cyclic.
superperfect group Yes It is the universal central extension of the perfect group alternating group:A5.

## Elements

Further information: element structure of special linear group:SL(2,5)

### Summary

Item Value
order of the whole group (total number of elements) 120
conjugacy class sizes 1,1,12,12,12,12,20,20,30
in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time)
maximum: 30, number of conjugacy classes: 9, lcm: 60
order statistics 1 of order 1, 1 of order 2, 20 of order 3, 30 of order 4, 24 of order 5, 20 of order 6, 24 of order 10
maximum: 10, lcm (exponent of the whole group): 60

## Subgroups

Further information: subgroup structure of special linear group:SL(2,5)

### Quick summary

Item Value
number of subgroups 76
number of conjugacy classes of subgroups 12
number of automorphism classes of subgroups 12
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers 2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5
3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10
5-Sylow: cyclic group:Z5, Sylow number 6
Hall subgroups Other than the whole group, trivial subgroup, and Sylow subgroups, there is a $\{ 2,3 \}$-Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no $\{ 2,5 \}$-Hall subgroup or $\{ 3,5 \}$-Hall subgroup.
maximal subgroups There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5).
normal subgroups The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5.

### Subgroup-defining functions

Subgroup-defining function What it means Value as subgroup Value as group Order Associated quotient-defining function Value as group Order (= index of subgroup)
center elements that commute with every group element center of special linear group:SL(2,5) cyclic group:Z2 2 inner automorphism group alternating group:A5 60
derived subgroup subgroup generated by all commutators whole group special linear group:SL(2,5) 120 abelianization trivial group 1
perfect core the subgroup at which the derived series stabilizes whole group special linear group:SL(2,5) 120  ? trivial group 1
hypocenter the subgroup at which the lower central series stabilizes whole group special linear group:SL(2,5) 120  ? trivial group 1
hypercenter the subgroup at which the upper central series stabilizes center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
Frattini subgroup intersection of all maximal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2 Frattini quotient alternating group:A5 60
Jacobson radical intersection of all maximal normal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
socle join of all minimal normal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2 socle quotient alternating group:A5 60
Baer norm intersection of normalizers of all subgroups center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
join of all abelian normal subgroups subgroup generated by all the abelian normal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
Fitting subgroup subgroup generated by all the nilpotent normal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
solvable radical subgroup generated by all the solvable normal subgroups center of special linear group:SL(2,5) cyclic group:Z2 2  ? alternating group:A5 60
socle over solvable radical its quotient by the solvable radical of the group is the socle of the quotient of the group by the solvable radical whole group special linear group:SL(2,5) 120  ? trivial group 1
permutation kernel too complicated to describe here whole group special linear group:SL(2,5) 120  ? trivial group 1

## Linear representation theory

Further information: linear representation theory of special linear group:SL(2,5)

### Summary

Item Value
Degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 1,2,2,3,3,4,4,5,6
maximum: 6, lcm: 60, number: 9, sum of squares: 120, quasirandom degree: 2

## GAP implementation

### Group ID

This finite group has order 120 and has ID 5 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,5)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(120,5);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [120,5]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

The group can be defined using GAP's SpecialLinearGroup function as:

Description Functions used
SL(2,5) or equivalently SpecialLinearGroup(2,5) SL
PerfectGroup(120) or equivalently PerfectGroup(120,1) PerfectGroup
SchurCover(AlternatingGroup(5)) SchurCover, AlternatingGroup