Projective special linear group:PSL(2,11)

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Definition

This group is defined as the projective special linear group of degree two over field:F11, the field with 11 elements.

Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	660	groups with same order	As $PSL(2,q)$, $q = 11$: $(q^3 - q)/2 = (11^3 - 11)/2 = 11(11 -1)(11 + 1)/2 = 11 \cdot 10 \cdot 12/2 = 660$
exponent of a group	330	groups with same order and exponent of a group | groups with same exponent of a group	As $PSL(2,q), q = 11$, underlying prime $p = 11$ (odd): $p(q^2 - 1)/4 = 11(11^2 - 1)/4 = 330$
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	the group is a simple non-abelian group
chief length	1	groups with same order and chief length | groups with same chief length	the group is a simple non-abelian group
composition length	1	groups with same order and composition length | groups with same composition length	the group is a simple non-abelian group

Arithmetic functions of an element-counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	8	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $PSL(2,q), q = 11$ ($q$ odd): $(q + 5)/2 = (11 + 5)/2 = 8$ See element structure of projective special linear group of degree two over a finite field, element structure of projective special linear group:PSL(2,11)
number of equivalence classes under rational conjugacy	6	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 projective special linear group:PSL(2,11)

Arithmetic functions of a subgroup-counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes of subgroups	16	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups	See subgroup structure of projective special linear group:PSL(2,11)
number of subgroups	620	groups with same order and number of subgroups | groups with same number of subgroups	See subgroup structure of projective special linear group:PSL(2,11)
number of automorphism classes of subgroups	14	groups with same order and number of automorphism classes of subgroups | groups with same number of automorphism classes of subgroups	See subgroup structure of projective special linear group:PSL(2,11)

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group, simple non-abelian group	Yes	projective special linear group is simple (with a couple of exceptions, but this isn't one of them)
minimal simple group	No	contains a subgroup isomorphic to alternating group:A5 (really?). See also classification of finite minimal simple groups

Subgroups

Further information: subgroup structure of projective special linear group:PSL(2,11)

Quick summary

Item	Value
Number of subgroups	620
Number of conjugacy classes of subgroups	16
Number of automorphism classes of subgroups	14
Isomorphism classes of Sylow subgroups and the corresponding fusion systems	2-Sylow: Klein four-group, fusion system is simple fusion system for Klein four-group, Sylow number is 55 3-Sylow: cyclic group:Z3, fusion system is the non-inner one, Sylow number is 55 5-Sylow: cyclic group:Z5, Sylow number is 66 11-Sylow: cyclic group:Z11, Sylow number is 12
Hall subgroups	Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{ 2,3 \}$-Hall subgroups (of order 12) and $\{ 2,3,5 \}$-Hall subgroups (of order 60), and $\{ 5,11 \}$-Hall subgroups (of order 55) Note that there are two distinct isomorphism classes of $\{ 2,3 \}$-Hall subgroups: dihedral group:D12 and alternating group:A4. This gives the smallest example illustrating that Hall not implies order-isomorphic
maximal subgroups	maximal subgroups have orders 12, 55, 60.
normal subgroups	only the whole group and the trivial subgroup, because the group is simple. See projective special linear group is simple (with a couple of small exceptions, but this isn't one of them)
subgroups that are simple non-abelian groups (apart from the whole group itself)	alternating group:A5 (order 60)

Linear representation theory

Further information: linear representation theory of projective special linear group:PSL(2,11)

Item	Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1,5,5,10,10,11,12,12 in grouped form: 1 (1 time), 5 (2 times), 10 (2 times), 11 (1 time), 12 (2 times) number: 8, sum of squares: 660, maximum: 12, quasirandom degree: 5, lcm: 660
number of irreducible representations (equals number of conjugacy classes)	8 As $PSL(2,q), q = 11$ ($q$ odd): $(q+ 5)/2 = (11 + 5)/2 = 8$

GAP implementation

Group ID

This finite group has order 660 and has ID 13 among the groups of order 660 in GAP's SmallGroup library. For context, there are 41 groups of order 660. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(660,13)

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

gap> G := SmallGroup(660,13);

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

IdGroup(G) = [660,13]

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(2,11)	PSL
PerfectGroup(660) or equivalently PerfectGroup(660,1)	PerfectGroup