Alternating group:A6

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Definition

The alternating group $A_6$ 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 $PSL(2,9)$, 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

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.

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 $\mathbb{Z}/6\mathbb{Z}$. 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

Character table

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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