Alternating group:A8
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Contents
Definition
This group is defined in the following equivalent ways:
- It is the alternating group of degree eight, i.e., over a set of size eight.
- It is the projective special linear group of degree four over the field of two elements, i.e., . It is also the special linear group , the projective general linear group , and the general linear group .
This is one member of the smallest order pair of non-isomorphic finite simple non-abelian groups having the same order. The other member of this pair is projective special linear group:PSL(3,4).
Equivalence of definitions
The equivalence between the various definitions within (2) follows from isomorphism between linear groups over field:F2.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 20160#Arithmetic functions
Basic arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 20160 | groups with same order | As alternating group : As general linear group : |
exponent of a group | 420 | groups with same order and exponent of a group | groups with same exponent of a group | As , even: As : |
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 all 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 |
Arithmetic functions of a counting nature
Function | Value | Similar groups | Explanation for function value |
---|---|---|---|
number of conjugacy classes | 14 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | As : (2 * (number of self-conjugate partitions of 8)) + (number of conjugate pairs of non-self-conjugate partitions of 8) = (more here) As : (more here) As : (more here) See element structure of alternating group:A8 |
number of conjugacy classes of subgroups | 137 | groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups | See subgroup structure of alternating group:A8, subgroup structure of alternating groups |
number of subgroups | 48337 | groups with same order and number of subgroups | groups with same number of subgroups | See subgroup structure of alternating group:A8, subgroup structure of alternating groups |
Elements
Further information: element structure of alternating group:A8
Subgroups
Further information: subgroup structure of alternating group:A8
Quick summary
Item | Value |
---|---|
Number of subgroups | 48337 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
Number of conjugacy classes of subgroups | 137 Compared with : 2, 5, 9, 22, 40, 137, ... |
Number of automorphism classes of subgroups | 112 Compared with : 2, 5, 9, 16, 37, 112, ... |
Isomorphism classes of Sylow subgroups | 2-Sylow: unitriangular matrix group:UT(4,2) (order 64). 3-Sylow: elementary abelian group:E9 (order 9). 5-Sylow: cyclic group:Z5 (order 5) 7-Sylow: cyclic group:Z7 (order 7) |
Hall subgroups | Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups of order 576. |
maximal subgroups | maximal subgroups have orders 360, 576, 720, 1344, 2520 |
normal subgroups | only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple. |
subgroups that are simple non-abelian groups (other than the whole group) | alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size , or via embedding it as in the alternating group on a set of size ), alternating group:A6 (order 360), alternating group:A7 (order 2520) |
Linear representation theory
Further information: linear representation theory of alternating group:A8
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1,7,14,20,21,21,21,28,35,45,45,56,64,70 grouped form: 1 (1 time), 7 (1 time) 14 (1 time), 20 (1 time), 21 (3 times), 28 (1 time), 35 (1 time), 45 (2 times), 56 (1 time), 64 (1 time), 70 (1 time) maximum: 70, lcm: 20160, number: 14, sum of squares: 20160 |
minimal splitting field, i.e., smallest field of realization of irreducible representations (characteristic zero) | where is a primitive seventh root of unity and is a primitive fifteenth root of unity Same as |
condition for a field of characteristic not 2,3,5,7 to be a splitting field | Both -7 and -15 should be squares in the field |
minimal splitting field, i.e., smallest field of realization of irreducible representations, prime characteristic not equal to 2,3,5,7 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
External links
GAP implementation
Description | Functions used |
---|---|
AlternatingGroup(8) | AlternatingGroup |
GL(4,2) | GL |
PGL(4,2) | PGL |
SL(4,2) | SL |
PSL(4,2) | PSL |
PerfectGroup(20160,4) | PerfectGroup |