Element structure of projective special linear group:PSL(3,2)
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This article gives specific information, namely, element structure, about a particular group, namely: projective special linear group:PSL(3,2).
View element structure of particular groups | View other specific information about projective special linear group:PSL(3,2)
This article describes the element structure of projective special linear group:PSL(3,2), which is the same as ,
. It is also isomorphic to the projective special linear group of degree two over field:F7, i.e., the group
.
Summary
Item | Value |
---|---|
order of the whole group (total number of elements) | 168 |
conjugacy class sizes | 1,21,24,24,42,56 in grouped form: 1 (1 time), 21 (1 time), 24 (2 times), 42 (1 time), 56 (1 time) maximum: 56, number of conjugacy classes: 6, lcm: 168 |
order statistics | 1 of order 1, 21 of order 2, 56 of order 3, 42 of order 4, 48 of order 7 maximum: 7, lcm (exponent of the whole group): 84 |
Conjugacy class structure
Interpretation as projective special linear group of degree two
Compare with element structure of projective special linear group of degree two over a finite field#Conjugacy class structure
We consider the group as ,
. We use the letter
to denote the generic case of
.
Nature of conjugacy class upstairs in ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class (generic ![]() |
Size of conjugacy class (![]() |
Number of such conjugacy classes (generic ![]() |
Number of such conjugacy classes (![]() |
Total number of elements (generic ![]() |
Total number of elements (![]() |
---|---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
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1 | 1 | 1 | 1 | 1 | 1 |
Diagonalizable over ![]() ![]() ![]() |
Square roots of ![]() |
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21 | 1 | 1 | ![]() |
21 |
Not diagonal, has Jordan block of size two | ![]() ![]() |
![]() ![]() |
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24 | 2 | 2 | ![]() |
48 |
Diagonalizable over ![]() ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() ![]() ![]() ![]() |
Same as characteristic polynomial | ![]() |
42 | ![]() |
1 | ![]() |
42 |
Diagonalizable over ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
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56 | ![]() |
1 | ![]() |
56 |
Total | NA | NA | NA | NA | NA | ![]() |
6 | ![]() |
168 |
Interpretation as general linear group of degree three over field:F2
Compare with element structure of general linear group of degree three#Conjugacy class structure. Since many cases (namely, those that rely on distinct eigenvalues) do not arise over field:F2, the cases have been omitted in the table below.
Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ![]() |
---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() |
![]() |
![]() |
1 | 1 | 1 | Yes | Yes |
Diagonalizable over ![]() ![]() |
Distinct Galois conjugate triple of elements in ![]() ![]() ![]() ![]() |
irreducible degree three polynomial over ![]() |
same as characteristic polynomial | 24 | 2 | 48 | Yes | No |
One eigenvalue is in ![]() ![]() |
one element of ![]() ![]() ![]() |
product of linear polynomial and irreducible degree two polynomial over ![]() |
same as characteristic polynomial | 56 | 1 | 56 | Yes | No |
Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over ![]() |
![]() ![]() |
![]() |
![]() |
42 | 1 | 42 | No | No |
Has Jordan block of size 3 | ![]() ![]() |
![]() |
same as characteristic polynomial | 21 | 1 | 21 | No | No |
Total | NA | NA | NA | NA | 6 | 168 | NA | NA |