# Difference between revisions of "Alternating group:A6"

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## Definition

The alternating group $A_6$ is defined in the following equivalent ways:

## 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

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 360 groups with same order As $A_n, n= 6$: $n!/2 = 6!/2 = 360$
As $PSL(2,q), q = 9$, $(q^3 - q)/\operatorname{gcd}(2,q-1) = (9^3 - 9)/2 = 360$
See order formulas for linear groups of degree two
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group As $A_n, n = 6$, $n$ even: $\operatorname{lcm} \{ 1,2,3,\dots,n - 1\} = \operatorname{lcm}\{1,2,3,4,5\} = 60$
As $PSL(2,q), q = 9, p = 3,$ ($p$ the underlying prime of $q$), $q$ odd: $p(q^2 - 1)/4 = 3(9^2 - 1)/4 = 60$
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
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 501 See subgroup structure of alternating group:A6
number of conjugacy classes 7 As $A_n, n = 6$: (2 * (number of self-conjugate partitions of 6)) + (number of conjugate pairs of non-self-conjugate partitions of 6) = $2(1) + 5 = 7$ (more here)
As $PSL(2,q)$, $q = 9$: $(q + 5)/2 = (9 + 5)/2 = 7$ (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 $(1,2,3)$, $(1,2,3,4,5)$ don't commute $A_n$ is non-abelian, $n \ge 4$.
Nilpotent group No Centerless: The center is trivial $A_n$ is non-nilpotent, $n \ge 4$.
Metacyclic group No Simple and non-abelian $A_n$ is not metacyclic, $n \ge 4$.
Supersolvable group No Simple and non-abelian $A_n$ is not supersolvable, $n \ge 4$.
Solvable group No $A_n$ is not solvable, $n \ge 5$.
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 $S_6$ gives outer automorphisms.
N-group Yes See classification of alternating groups that are N-groups $A_n$ is a N-group only for $n \le 7$.

## 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 $A_n, n = 3,4,5,\dots$: 2, 10, 59, 501, 3786, 48337, ...
number of conjugacy classes of subgroups 22
Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 22, 40, 137, ...
number of automorphism classes of subgroups 16
Compared with $A_n, n = 3,4,5,\dots$: 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 $\{ 2,3 \}$-Hall subgroup, $\{ 2,5 \}$-Hall subgroup, and $\{ 3,5 \}$-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 $A_6$ 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 $\{ (), (1,2)(3,4) \}$ cyclic group:Z2 2 180 1 45 45
V4 with two fixed points in A6 $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ Klein four-group 4 90 1 15 15
V4 without fixed points in A6 $\{ (), (1,2)(5,6), (1,2)(3,4), (3,4)(5,6) \}$ Klein four-group 4 90 1 15 15
Z4 in A6 $\{ (), (1,2,3,4)(5,6), (1,3)(2,4), (1,4,3,2)(5,6) \}$ cyclic group:Z4 4 90 1 45 45
D8 in A6 $\langle (1,2,3,4)(5,6), (1,3)(5,6) \rangle$ dihedral group:D8 8 45 1 45 45 2-Sylow
A3 in A6 $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 120 1 20 20
diagonal A3 in A6 $\{ (), (1,2,3)(4,5,6), (1,3,2)(4,6,5) \}$ cyclic group:Z3 3 120 1 20 20
E9 in A6 $\langle (1,2,3), (4,5,6) \rangle$ elementary abelian group:E9 9 40 1 10 10 3-Sylow; also maximal among $\{ 3,5 \}$-subgroups
diagonal S3 in A6 $\langle (1,2,3)(4,5,6), (1,2)(4,5)\rangle$ symmetric group:S3 6 60 1 60 60
twisted S3 in A6 $\langle (1,2,3), (1,2)(4,5) \rangle$ symmetric group:S3 6 60 1 60 60
A4 in A6 $\langle (1,2)(3,4), (1,2,3) \rangle$ alternating group:A4 12 30 1 15 15
twisted A4 in A6 $\langle (1,2,3)(4,5,6), (1,4)(2,5), (1,4)(3,6) \rangle$ alternating group:A4 12 30 1 15 15
standard twisted S4 in A6 $\langle (1,2,3,4)(5,6), (1,2)(5,6) \rangle$ symmetric group:S4 24 15 1 15 15 maximal; also maximal among $\{ 2,3 \}$-subgroups
exceptional twisted S4 in A6 $\langle (3,4)(5,6), (1,2)(5,6), (1,3,5)(2,4,6), (3,5)(4,6) \rangle$ symmetric group:S4 24 15 1 15 15 maximal; also maximal among $\{ 2,3 \}$-subgroups
generalized dihedral group for E9 in A6 $\langle (1,2,3), (4,5,6), (1,2)(4,5) \rangle$ generalized dihedral group for E9 18 20 1 10 10
? $\langle (4,5,6), (1,2,3), (2,3)(5,6), (1,4)(2,5,3,6) \rangle$  ? 36 10 1 10 10 maximal; also maximal among $\{ 2,3 \}$-subgroups
Z5 in A6 $\langle (1,2,3,4,5) \rangle$ cyclic group:Z5 5 72 1 36 36 5-Sylow, also maximal among $\{ 3,5 \}$-subgroups
D10 in A6 $\langle (1,2,3,4,5), (2,5)(3,4) \rangle$ dihedral group:D10 10 36 1 36 36 maximal among $\{ 2,5 \}$-subgroups
A5 in A6 $\langle (1,2,3,4,5), (1,2,3) \rangle$ alternating group:A5 60 6 1 6 6 maximal
twisted A5 in A6 $\langle (1,2,3,4,5), (1,4)(5,6) \rangle$ alternating group:A5 60 6 1 6 6 maximal
whole group $\langle (1,2,3,4,5), (1,2,3), (1,2)(5,6) \rangle$ alternating group:A6 360 1 1 1 1
Total -- -- -- -- 22 -- 501 --

### Table classifying subgroups up to automorphisms

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 $\{ (), (1,2)(3,4) \}$ cyclic group:Z2 2 180 1 45 45
V4 with two fixed points in A6, V4 without fixed points in A6 $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$
$\{ (), (1,2)(5,6), (1,2)(3,4), (3,4)(5,6) \}$
Klein four-group 4 90 2 15 30
Z4 in A6 $\{ (), (1,2,3,4)(5,6), (1,3)(2,4), (1,4,3,2)(5,6) \}$ cyclic group:Z4 4 90 1 45 45
D8 in A6 $\langle (1,2,3,4)(5,6), (1,3)(5,6) \rangle$ dihedral group:D8 8 45 1 45 45 2-Sylow
A3 in A6
diagonal A3 in A6
$\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (1,2,3)(4,5,6), (1,3,2)(4,6,5) \}$
cyclic group:Z3 3 120 2 20 40
E9 in A6 $\langle (1,2,3), (4,5,6) \rangle$ elementary abelian group:E9 9 40 1 10 10 3-Sylow
diagonal S3 in A6
twisted S3 in A6
$\langle (1,2,3)(4,5,6), (1,2)(4,5)\rangle$
$\langle (1,2,3), (1,2)(4,5) \rangle$
symmetric group:S3 6 60 2 60 120
A4 in A6
twisted A4 in A6
$\langle (1,2)(3,4), (1,2,3) \rangle$
$\langle (1,2,3)(4,5,6), (1,4)(2,5), (1,4)(3,6) \rangle$
alternating group:A4 12 30 2 15 30
standard twisted S4 in A6, exceptional twisted S4 in A6 $\langle (1,2,3,4)(5,6), (1,2)(5,6) \rangle$, $\langle (3,4)(5,6), (1,2)(5,6), (1,3,5)(2,4,6), (3,5)(4,6) \rangle$ symmetric group:S4 24 15 2 15 30 maximal
generalized dihedral group for E9 in A6 $\langle (1,2,3), (4,5,6), (1,2)(4,5) \rangle$ generalized dihedral group for E9 18 20 1 10 10
? $\langle (4,5,6), (1,2,3), (2,3)(5,6), (1,4)(2,5,3,6) \rangle$  ? 36 10 1 10 10 maximal
Z5 in A6 $\langle (1,2,3,4,5) \rangle$ cyclic group:Z5 5 72 1 36 36 5-Sylow
D10 in A6 $\langle (1,2,3,4,5), (2,5)(3,4) \rangle$ dihedral group:D10 10 36 1 36 36
A5 in A6
twisted A5 in A6
$\langle (1,2,3,4,5), (1,2,3) \rangle$
$\langle (1,2,3,4,5), (1,4)(5,6) \rangle$
alternating group:A5 60 6 2 6 12 maximal
whole group $\langle (1,2,3,4,5), (1,2,3), (1,2)(5,6) \rangle$ alternating group:A6 360 1 1 1 1
Total -- -- -- -- 22 -- 501 --

## 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 $A_6$ as a self-centralizing normal subgroup. The whole automorphism group contains $A_6$ as a NSCFN-subgroup.

Below is the complete list of these groups.:

Group containing $A_6 = \operatorname{Inn}(A_6)$ and contained in $\operatorname{Aut}(A_6)$ (this is an almost simple group) Corresponding subgroup of $\operatorname{Out}(A_6)$ viewed as Klein four-group Order of group Order of corresponding subgroup of $\operatorname{Out}(A_6)$ = index of $A_6$ 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.

The Schur multiplier of alternating group:A6 is cyclic group:Z6, i.e., the group $\mathbb{Z}/6\mathbb{Z}$. For each of the possible quotient groups of $\mathbb{Z}/6\mathbb{Z}$, 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 $A_6$ 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 $2 \cdot A_6$ 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 $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 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 $\mathbb{Z}[2\cos(2\pi/5)] = \mathbb{Z}[(1 + \sqrt{5})/2]$
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) $\mathbb{Q}(\sqrt{5})$
Quadratic extension of $\mathbb{Q}$
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 $S_6$ 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 $\sqrt{5} \mapsto -\sqrt{5}$)
Minimal splitting field in prime characteristic $p \ne 2,3,5$ Case $p \equiv \pm 1 \pmod 5$: prime field $\mathbb{F}_p$
Case $p \equiv \pm 2 \pmod 5$: quadratic extension $\mathbb{F}_{p^2}$ of $\mathbb{F}_p$
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 $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 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 $\mathbb{Z}[2\cos(2\pi/5)] = \mathbb{Z}[(1 + \sqrt{5})/2]$
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) $\mathbb{Q}(\sqrt{5})$
Quadratic extension of $\mathbb{Q}$
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 $S_6$ 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 $\sqrt{5} \mapsto -\sqrt{5}$)
Minimal splitting field in prime characteristic $p \ne 2,3,5$ Case $p \equiv \pm 1 \pmod 5$: prime field $\mathbb{F}_p$
Case $p \equiv \pm 2 \pmod 5$: quadratic extension $\mathbb{F}_{p^2}$ of $\mathbb{F}_p$
Smallest size splitting field field:F11
Degrees of irreducible representations over the rational numbers 1,5,5,9,10,16

## 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 $G$:

gap> G := SmallGroup(360,118);

Conversely, to check whether a given group $G$ 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