Special linear group:SL(2,3)

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Definition

The special linear group $SL(2,3)$ is defined in the following equivalent ways:

• It is the special linear group of degree two over a field of three elements. In other words, it is the group of $2\times 2$ matrices with determinant $1$ over the field of three elements.
• It is the double cover of alternating group $2\cdot A_4$, i.e., it is the double cover of alternating group:A4.
• It is a binary von Dyck group with the parameters $(2,3,3)$, i.e., it has the presentation:

$⟨a,b,c\mid a^3=b^3=c^2=abc⟩$.

It is a group of order $24$.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	24	groups with same order	As $SL(2,q)$, $q=3$: $q^3-q=3^3-3=24$ As $2\cdot A_n,n=4$: $n!=4!=24$ As binary von Dyck group with parameters $(p,q,r)=(2,3,3)$: $\frac{4}{1/p+1/q+1/r-1}=\frac{4}{1/2+1/3+1/3-1}=\frac{4}{1/6}=24$
exponent of a group	12	groups with same order and exponent of a group | groups with same exponent of a group	Elements of order $2,3,4,6$.
derived length	3	groups with same order and derived length | groups with same derived length	Derived series goes through quaternion group and then the center.
nilpotency class	--	--	not a nilpotent group.
Frattini length	2	groups with same order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	Element of order three and element of order four.
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	4	groups with same order and max-length of a group | groups with same max-length of a group	Series for Sylow 2-subgroup followed by whole group, OR derived series refined at end.
composition length	4	groups with same order and composition length | groups with same composition length
chief length	3	groups with same order and chief length | groups with same chief length

Arithmetic functions of an element-counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	7	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $SL(2,q),q=3$ (odd): $q+4=3+4=7$ As binary von Dyck group with parameters $(p,q,r)=(2,3,3)$: $p+q+r-1=2+3+3-1=7$ As $2\cdot A_n,n=4$: (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) $=5+3(1)-1=7$ See element structure of special linear group:SL(2,3)
number of equivalence classes under rational conjugacy	5	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(2,3)
number of conjugacy classes of rational elements	3	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(2,3)

Arithmetic functions of a subgroup-counting nature

Function	Value	Similar groups	Explanation for function value
number of subgroups	15	groups with same order and number of subgroups | groups with same number of subgroups
number of conjugacy classes of subgroups	7	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

Group properties

Important properties

Property	Satisfied?	Explanation
Abelian group	No
Nilpotent group	No
Metacyclic group	No
Supersolvable group	No
Solvable group	Yes	Length three, commutator subgroup is the quaternion group, next commutator subgroup is center.

Other properties

Property	Satisfied?	Explanation
T-group	No
Monolithic group	Yes	The center of order two is the unique minimal normal subgroup.
One-headed group	Yes	The commutator subgroup, which is a quaternion group
Group having subgroups of all orders dividing the group order	No
Schur-trivial group	Yes	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Elements

Further information: element structure of special linear group:SL(2,3)

Summary

Item	Value
order of the whole group (total number of elements)	24
conjugacy class sizes	1,1,4,4,4,4,6 grouped form: 1 (2 times), 4 (4 times), 6 (1 time) maximum: 6, number of conjugacy classes: 7, lcm: 12
order statistics	1 of order 1, 1 of order 2, 8 of order 3, 6 of order 4, 8 of order 6 maximum: 6, lcm (exponent of the whole group): 12

Conjugacy classes

Note that since we are over field:F3, $-1=2$, so all the $-1$s below can be rewritten as $2$s.

COMBINATORIAL BREAKDOWN TABLE: The table below breaks down a collection into various classes or types and provides information on the counts for each type. For some of the columns, totals provide a sanity check that all elements or classes have been accounted for. In this case, the table gives information on conjugacy class structure.

Conjugacy class representative	Conjugacy class size	List of all elements of conjugacy class	Order of elements in conjugacy class
$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1
$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	2
$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ -1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& -1\\ \end{array}\right)$	3
$\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& -1\\ \end{array}\right)$	3
$\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 1\\ \end{array}\right)$	6
$\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& 1\\ \end{array}\right)$	6
$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	6	[SHOW MORE] $\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ -1& 1\\ \end{array}\right)$	4
Total	24 (order of the group)	--	--

Automorphism classes

COMBINATORIAL BREAKDOWN TABLE: The table below breaks down a collection into various classes or types and provides information on the counts for each type. For some of the columns, totals provide a sanity check that all elements or classes have been accounted for. In this case, the table gives information on automorphism class structure.

Below are the orbits under the action of the automorphism group, i.e., the automorphism classes of elements of the group.

List of representatives for each conjugacy class in the automorphism class	Number of conjugacy classes in the automorphism class	Size of each conjugacy class	Automorphism class size	Order of elements in conjugacy class
$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1	1	1	1
$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	1	1	1	2
$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$	2	4	8	3
$\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$	2	4	8	6
$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	1	6	6	4
Total	7 (number of conjugacy classes)	--	24 (order of the group)	--

Endomorphisms

Automorphisms

The automorphism group of $SL(2,3)$ is isomorphic to the symmetric group of degree four. The inner automorphism group, which is the quotient of $SL(2,3)$ by its center, and is also the projective special linear group $PSL(2,3)$, is isomorphic to the alternating group of degree four.

To see how the outer automorphisms act, we can view $SL(2,3)$ as a subgroup of index two in $GL(2,3)$, the general linear group of order two over the field of three elements. The inner automorphism group, which is $PGL(2,3)$, is isomorphic to the symmetric group of degree four. Since $SL(2,3)$ is a normal subgroup, and since its center equals the center of $GL(2,3)$, the automorphisms of $GL(2,3)$ all restrict to automorphisms of $SL(2,3)$. $SL(2,3)$ has no other automorphisms.

Subgroups

Further information: Subgroup structure of special linear group:SL(2,3)

Quick summary

Item	Value
number of subgroups	15
number of conjugacy classes of subgroups	7
number of automorphism classes of subgroups	7
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers	2-Sylow: quaternion group (order 8) as Q8 in SL(2,3) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 1 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 4
Hall subgroups	The order has only two prime divisors, so no possibility of Hall subgroups other than trivial subgroup, whole group, and Sylow subgroups
maximal subgroups	There are maximal subgroups of orders 6 (Z6 in SL(2,3)) and 8 (2-Sylow subgroup of special linear group:SL(2,3))
normal subgroups	There are two proper nontrivial normal subgroups: center of special linear group:SL(2,3) and 2-Sylow subgroup of special linear group:SL(2,3)

Table classifying subgroups up to automorphism

Note that, in the matrices, -1 can be written as 2 since elements are taken modulo 3.

Automorphism class of subgroups	Representative subgroup	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
trivial subgroup	$\{\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)\}$	trivial group	1	24	1	1	1	special linear group:SL(2,3)	1	trivial
center of special linear group:SL(2,3)	$\{\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right),\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)\}$	cyclic group:Z2	2	12	1	1	1	alternating group:A4	1
Z4 in SL(2,3)	$⟨\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)⟩$	cyclic group:Z4	4	6	1	3	3	--	2
2-Sylow subgroup of special linear group:SL(2,3)	$⟨\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right),\left(\begin{array}{cc}-1& -1\\ -1& 1\\ \end{array}\right)⟩$	quaternion group	8	3	1	1	1	cyclic group:Z3	1	2-Sylow
Z3 in SL(2,3)	$⟨\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)⟩$	cyclic group:Z3	3	8	1	4	4	--	--	3-Sylow
Z6 in SL(2,3)	$⟨\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right),\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)⟩$	cyclic group:Z6	6	4	1	4	4	--	--	3-Sylow normalizer
whole group	all elements	special linear group:SL(2,3)	24	1	1	1	1	trivial group	0
Total (7 rows)	--	--	--	--	7	--	15	--	--	--

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(2,3)	cyclic group:Z2	2	inner automorphism group	alternating group:A4	12
derived subgroup	subgroup generated by all commutators	2-Sylow subgroup of special linear group:SL(2,3)	quaternion group	8	abelianization	cyclic group:Z3	3
second derived subgroup	derived subgroup of derived subgroup	center of special linear group:SL(2,3)	cyclic group:Z2	2	?	alternating group:A4	12
perfect core	the subgroup at which the derived series stabilizes	trivial subgroup	trivial group	1	?	special linear group:SL(2,3)	24
hypocenter	the subgroup at which the lower central series stabilizes	2-Sylow subgroup of special linear group:SL(2,3)	quaternion group	8	?	cyclic group:Z3	3
hypercenter	the subgroup at which the upper central series stabilizes	center of special linear group:SL(2,3)	cyclic group:Z2	2	?	alternating group:A4	12
Frattini subgroup	intersection of all maximal subgroups	center of special linear group:SL(2,3)	cyclic group:Z2	2	Frattini quotient	alternating group:A4	12
Jacobson radical	intersection of all maximal normal subgroups	2-Sylow subgroup of special linear group:SL(2,3)	quaternion group	8	?	cyclic group:Z3	1
socle	join of all minimal normal subgroups	center of special linear group:SL(2,3)	cyclic group:Z2	2	socle quotient	alternating group:A4	12
Baer norm	intersection of normalizers of all subgroups	center of special linear group:SL(2,3)	cyclic group:Z2	2	?	alternating group:A4	12
join of all abelian normal subgroups	subgroup generated by all the abelian normal subgroups	center of special linear group:SL(2,3)	cyclic group:Z2	2	?	alternating group:A4	12
Fitting subgroup	subgroup generated by all the nilpotent normal subgroups	2-Sylow subgroup of special linear group:SL(2,3)	quaternion group	8	?	cyclic group:Z3	1
solvable radical	subgroup generated by all the solvable normal subgroups	whole group	special linear group:SL(2,3)	24	?	trivial group	1

Supergroups

The special linear group $SL(2,3)$ is a subgroup of the general linear group $GL(2,3)$. It is also a subgroup of $SL(2,q)$ for any prime power $q\ge 3$ (note that $q$ need not be a power of $3$). For full proof, refer: SL(2,3) is a subgroup of any special linear group, Dickson's theorem

GAP implementation

Group ID

This finite group has order 24 and has ID 3 among the groups of order 24 in GAP's SmallGroup library. For context, there are groups of order 24. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(24,3)

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

gap> G := SmallGroup(24,3);

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

IdGroup(G) = [24,3]

or just do:

IdGroup(G)

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

Other descriptions

Description	Functions used
SL(2,3) or SpecialLinearGroup(2,3)	SL
SchurCover(AlternatingGroup(4))	SchurCover, AlternatingGroup

Description by presentation

F := FreeGroup(3);
G := F/[F.1^3*F.2^(-3), F.2^3*F.3^(-2), F.1^3*(F.1*F.2*F.3)^(-1)];