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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.
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
|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: |
|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|
Further information: element structure of alternating group:A8
Further information: subgroup structure of alternating group:A8
|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
|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|
|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]|