Groupprops, The Group Properties Wiki (pre-alpha)

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This article is about a particular group, viz a group unique upto isomorphism[SHOW MORE]

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Definition

This group is defined in many equivalent ways:

• It is the projective special linear group of degree three over the field of two elements, i.e., PSL(3,2).
• It is the special linear group of degree three over the field of two elements, i.e., SL(3,2).
• It is the projective general linear group of degree three over the field of two elements, i.e., PGL(3,2).
• It is the general linear group of degree three over the field of two elements, i.e., GL(3,2).
• It is the projective special linear group of degree two over the field of seven elements, i.e., PSL(2,7).

Arithmetic functions

Function	Value	Explanation
order	168
exponent	84	Elements of order 2,3,4,7.
derived length	--	not a solvable group.
nilpotency class	--	not a nilpotent group.
Frattini length	1	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2	Generated by an element of order 2 and an element of order 3.
subgroup rank	2	--
max-length	5
number of subgroups	179
number of conjugacy classes	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