Alternating group:A6
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Contents
Definition
The alternating group is defined in the following equivalent ways:
- It is the group of even permutations (viz., the alternating group) on six elements.
- It is the projective special linear group , i.e., the projective special linear group of degree two over field:F9.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 360#Arithmetic functions
Basic arithmetic functions
Arithmetic functions of a counting nature
Function | Value | Explanation |
---|---|---|
number of subgroups | 501 | See subgroup structure of alternating group:A6 |
number of conjugacy classes | 7 | As : (2 * (number of self-conjugate partitions of 6)) + (number of conjugate pairs of non-self-conjugate partitions of 6) = (more here) As , : (more here) See element structure of alternating group:A6 |
number of conjugacy classes of subgroups | 22 | See subgroup structure of alternating group:A6 |
Group properties
Property | Satisfied | Explanation | Comment |
---|---|---|---|
Abelian group | No | , don't commute | is non-abelian, . |
Nilpotent group | No | Centerless: The center is trivial | is non-nilpotent, . |
Metacyclic group | No | Simple and non-abelian | is not metacyclic, . |
Supersolvable group | No | Simple and non-abelian | is not supersolvable, . |
Solvable group | No | is not solvable, . | |
Simple non-abelian group | Yes | alternating groups are simple, projective special linear group is simple | |
T-group | Yes | Simple and non-abelian | |
Ambivalent group | Yes | Classification of ambivalent alternating groups | |
Rational-representation group | No | ||
Rational group | No | ||
Complete group | No | Conjugation by odd permutations in gives outer automorphisms. | |
N-group | Yes | See classification of alternating groups that are N-groups | is a N-group only for . |
Elements
Further information: element structure of alternating group:A6
Subgroups
Further information: subgroup structure of alternating group:A6
Quick summary
Item | Value |
---|---|
number of subgroups | 501 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
number of conjugacy classes of subgroups | 22 Compared with : 2, 5, 9, 22, 40, 137, ... |
number of automorphism classes of subgroups | 16 Compared with : 2, 5, 9, 16, 37, 112, ... |
isomorphism classes of Sylow subgroups and the corresponding fusion systems | 2-Sylow: dihedral group:D8 (order 8) as D8 in A6 (with its simple fusion system -- see simple fusion system for dihedral group:D8). Sylow number is 45. 3-Sylow: elementary abelian group:E9 (order 9) as E9 in A6. Sylow number is 10. 5-Sylow: cyclic group:Z5 (order 5) as Z5 in A6. Sylow number is 36. |
Hall subgroups | no Hall subgroups other than the Sylow subgroups, whole group, and trivial subgroup. In particular, there is no -Hall subgroup, -Hall subgroup, and -Hall subgroup. |
maximal subgroups | maximal subgroups have orders 24, 36, and 60. |
normal subgroups | only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple. |
Table classifying subgroups up to permutation automorphisms
Note that alternating groups are simple (with an exception for degree 1,2,4), so in particular is simple. Hence no proper nontrivial subgroup is normal or subnormal.
The below lists subgroups up to automorphisms arising from permutations, i.e., automorphisms arising from conjugation in symmetric group:S6. This is not the same as the classification up to automorphisms because of the presence of other automorphisms, a phenomenon unique to degree six.
TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.
Permutation automorphism class of subgroups | Representative subgroup (full list if small, generating set if large) | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Total number of subgroups | Note |
---|---|---|---|---|---|---|---|---|
trivial subgroup | trivial group | 1 | 360 | 1 | 1 | 1 | trivial | |
subgroup generated by double transposition in A6 | cyclic group:Z2 | 2 | 180 | 1 | 45 | 45 | ||
V4 with two fixed points in A6 | Klein four-group | 4 | 90 | 1 | 15 | 15 | ||
V4 without fixed points in A6 | Klein four-group | 4 | 90 | 1 | 15 | 15 | ||
Z4 in A6 | cyclic group:Z4 | 4 | 90 | 1 | 45 | 45 | ||
D8 in A6 | dihedral group:D8 | 8 | 45 | 1 | 45 | 45 | 2-Sylow | |
A3 in A6 | cyclic group:Z3 | 3 | 120 | 1 | 20 | 20 | ||
diagonal A3 in A6 | cyclic group:Z3 | 3 | 120 | 1 | 20 | 20 | ||
E9 in A6 | elementary abelian group:E9 | 9 | 40 | 1 | 10 | 10 | 3-Sylow; also maximal among -subgroups | |
diagonal S3 in A6 | symmetric group:S3 | 6 | 60 | 1 | 60 | 60 | ||
twisted S3 in A6 | symmetric group:S3 | 6 | 60 | 1 | 60 | 60 | ||
A4 in A6 | alternating group:A4 | 12 | 30 | 1 | 15 | 15 | ||
twisted A4 in A6 | alternating group:A4 | 12 | 30 | 1 | 15 | 15 | ||
standard twisted S4 in A6 | symmetric group:S4 | 24 | 15 | 1 | 15 | 15 | maximal; also maximal among -subgroups | |
exceptional twisted S4 in A6 | symmetric group:S4 | 24 | 15 | 1 | 15 | 15 | maximal; also maximal among -subgroups | |
generalized dihedral group for E9 in A6 | generalized dihedral group for E9 | 18 | 20 | 1 | 10 | 10 | ||
? | ? | 36 | 10 | 1 | 10 | 10 | maximal; also maximal among -subgroups | |
Z5 in A6 | cyclic group:Z5 | 5 | 72 | 1 | 36 | 36 | 5-Sylow, also maximal among -subgroups | |
D10 in A6 | dihedral group:D10 | 10 | 36 | 1 | 36 | 36 | maximal among -subgroups | |
A5 in A6 | alternating group:A5 | 60 | 6 | 1 | 6 | 6 | maximal | |
twisted A5 in A6 | alternating group:A5 | 60 | 6 | 1 | 6 | 6 | maximal | |
whole group | alternating group:A6 | 360 | 1 | 1 | 1 | 1 | ||
Total | -- | -- | -- | -- | 22 | -- | 501 | -- |
Table classifying subgroups up to automorphisms
Supergroups
Further information: supergroups of alternating group:A6
Subgroups: making some or all the outer automorphisms inner
All the groups listed here are almost simple groups, because alternating group:A6 is a simple non-abelian group.
The outer automorphism group of alternating group:A6 is a Klein four-group. In particular, it has order 4. By the fourth isomorphism theorem, subgroups of the automorphism group containing the inner automorphism group correspond to subgroups of the outer automorphism group, which is the quotient group of the automorphism group by the inner automorphism group.
Since alternating group:A6 is a centerless group, it is identified naturally with its inner automorphism group, so each of the subgroups of the automorphism group containing the inner automorphism group is also a group containing as a self-centralizing normal subgroup. The whole automorphism group contains as a NSCFN-subgroup.
Below is the complete list of these groups.:
Group containing and contained in (this is an almost simple group) | Corresponding subgroup of viewed as Klein four-group | Order of group | Order of corresponding subgroup of = index of in group = order of group/360 | Second part of GAP ID of big group (GAP ID is (order,2nd part)) |
---|---|---|---|---|
alternating group:A6 | trivial subgroup | 360 | 1 | 118 |
symmetric group:S6 | one of the three copies of Z2 in V4 | 720 | 2 | 763 |
projective general linear group:PGL(2,9) | one of the three copies of Z2 in V4 | 720 | 2 | 764 |
Mathieu group:M10 | one of the three copies of Z2 in V4 | 720 | 2 | 765 |
automorphism group of alternating group:A6 | whole group | 1440 | 4 | 5841 |
Quotients: Stem extensions and Schur covering groups
All the groups listed here are quasisimple groups, because alternating group:A6 is a simple non-abelian group.
Further information: Group cohomology of alternating group:A6#Schur multiplier, group cohomology of alternating groups
The Schur multiplier of alternating group:A6 is cyclic group:Z6, i.e., the group . For each of the possible quotient groups of , there is a unique stem extension with that as base normal subgroup and alternating group:A6 as quotient. The stem extension for the whole Schur multiplier is the unique Schur covering group, also called the universal central extension.
Note that uniqueness follows from being a perfect group.
The list is below:
Group at base of stem extension | Order | Corresponding stem extension group (this is a quasisimple group) | Order | Second part of GAP ID (GAP ID is (order,second part)) |
---|---|---|---|---|
trivial group | 1 | alternating group:A6 | 360 | 118 |
cyclic group:Z2 | 2 | special linear group:SL(2,9), also denoted to indicate that it is a double cover of alternating group | 720 | 409 |
cyclic group:Z3 | 3 | triple cover of alternating group:A6 | 1080 | 260 |
cyclic group:Z6 | 6 | Schur cover of alternating group:A6 | 2160 | (ID not available for this order) |
Linear representation theory
Further information: linear representation theory of alternating group:A6
Item | Value |
---|---|
Degrees of irreducible representations over a splitting field (such as or ) | 1,5,5,8,8,9,10 grouped form: 1 (1 time), 5 (2 times), 8 (2 times), 9 (1 time), 10 (1 time) maximum: 10, lcm: 360, number: 7, sum of squares: 360 |
Ring generated by character values | |
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) | Quadratic extension of Same as field generated by character values |
Orbits of irreducible representations under action of automorphism group | orbits of size 1 for representations of degree 1,9,10; orbits of size two for degree 5 and degree 8 representations (the degree 8 representations are interchanged under conjugation by an odd permutation; the degree 5 representations are interchaged by an automorphism that is outer for as well) |
Orbits of irreducible representations under action of Galois group | orbits of size 1 for representations of degree 1,5,5,9,10; orbit of size two for degree 8 representations (automorphism ) |
Minimal splitting field in prime characteristic | Case : prime field Case : quadratic extension of |
Smallest size splitting field | field:F11 |
Degrees of irreducible representations over the rational numbers | 1,5,5,9,10,16 |
Summary
Item | Value |
---|---|
Degrees of irreducible representations over a splitting field (such as or ) | 1,5,5,8,8,9,10 grouped form: 1 (1 time), 5 (2 times), 8 (2 times), 9 (1 time), 10 (1 time) maximum: 10, lcm: 360, number: 7, sum of squares: 360 |
Ring generated by character values | |
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) | Quadratic extension of Same as field generated by character values |
Orbits of irreducible representations under action of automorphism group | orbits of size 1 for representations of degree 1,9,10; orbits of size two for degree 5 and degree 8 representations (the degree 8 representations are interchanged under conjugation by an odd permutation; the degree 5 representations are interchaged by an automorphism that is outer for as well) |
Orbits of irreducible representations under action of Galois group | orbits of size 1 for representations of degree 1,5,5,9,10; orbit of size two for degree 8 representations (automorphism ) |
Minimal splitting field in prime characteristic | Case : prime field Case : quadratic extension of |
Smallest size splitting field | field:F11 |
Degrees of irreducible representations over the rational numbers | 1,5,5,9,10,16 |
Character table
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]GAP implementation
Group ID
This finite group has order 360 and has ID 118 among the groups of order 360 in GAP's SmallGroup library. For context, there are 162 groups of order 360. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(360,118)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(360,118);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [360,118]
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 |
---|---|
AlternatingGroup(6) | AlternatingGroup |
PSL(2,9) | PSL |