Projective special linear group:PSL(3,2)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

This group is defined in many equivalent ways:

1. It is the projective special linear group of degree three over the field of two elements, i.e., $PSL(3,2)$.
2. It is the special linear group of degree three over the field of two elements, i.e., $SL(3,2)$.
3. It is the projective general linear group of degree three over the field of two elements, i.e., $PGL(3,2)$.
4. It is the general linear group of degree three over the field of two elements, i.e., $GL(3,2)$.
5. It is the projective special linear group of degree two over the field of seven elements, i.e., $PSL(2,7)$.
6. It is the conformal automorphism group of the Klein quartic surface, which is a Riemann surface and in particular a Hurwitz surface. Hence, this group is a Hurwitz group, and is in fact the unique Hurwitz group of smallest order.

Equivalence of definitions

The equivalence between definitions (1)-(4) follows from isomorphism between linear groups over field:F2.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	168	groups with same order	As $\! GL(n,q), n = 3, q = 2$: $(q^n - 1)(q^n - q) \dots (q^n - q^{n-1}) = (2^3 - 1)(2^3 - 2)(2^3 - 2^2) = 7 \cdot 6 \cdot 4 = 168$ As $\! PSL(2,q), q = 7$: $\! (q^3 - q)/2 = q(q - 1)(q + 1)/2 = (7)(6)(8)/2 = 168$
exponent of a group	84	groups with same order and exponent of a group | groups with same exponent of a group	Elements of order $2,3,4,7$.
derived length	--		not a solvable group.
nilpotency class	--		not a nilpotent group.
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	Generated by an element of order $2$ and an element of order $3$.
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

Arithmetic functions of a counting nature

Function	Value	Explanation
number of subgroups	179
number of conjugacy classes	6	As $PSL(2,q)$, $q = 7$: $(q + 5)/2 = (7 + 5)/2 = 6$ As $GL(3,q), q = 2$: $q^3 - q = 2^3 - 2 = 6$ As $SL(3,q), q = 2$: $q^2 + q = 2^2 + 2 = 6$
number of conjugacy classes of subgroups	15

Group properties

Property	Satisfied	Explanation
Abelian group	No
Nilpotent group	No
Metacyclic group	No
Supersolvable group	No
Solvable group	No
Simple group	Yes	The second smallest simple non-abelian group (hence finite simple non-abelian group).
T-group	Yes
HN-group	No
Monolithic group	Yes
One-headed group	Yes
Hurwitz group	Yes	It is the conformal automorphism group of the Klein quartic surface, which has genus three, and its order is $84(3 - 1) = 168$.

GAP implementation

Group ID

This finite group has order 168 and has ID 42 among the groups of order 168 in GAP's SmallGroup library. For context, there are 57 groups of order 168. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(168,42)

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

gap> G := SmallGroup(168,42);

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

IdGroup(G) = [168,42]

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
PSL(3,2)	PSL
PGL(3,2)	PGL
SL(3,2)	SL
GL(3,2)	GL
PSL(2,7)	PSL
PSU(2,7)	PSU