# Element structure of general linear group:GL(2,3)

From Groupprops

This article gives specific information, namely, element structure, about a particular group, namely: general linear group:GL(2,3).

View element structure of particular groups | View other specific information about general linear group:GL(2,3)

This article describes the element structure of general linear group:GL(2,3) which is the general linear group of degree two over field:F3.

See also element structure of general linear group of degree two.

## Conjugacy class structure

### Interpretation as general linear group of degree two over field:F3

Compare with element structure of general linear group of degree two#Conjugacy class structure.

Nature of conjugacy class | 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 () | Representative matrix (one per conjugacy class) |
---|---|---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | where | where | where | 1 | 1 | 2 | 2 | , | ||

Diagonalizable over field:F9, not over . Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of | , , | Same as characteristic polynomial | 6 | 3 | 18 | , , | |||

Not diagonal, has Jordan block of size two | (multiplicity two) where | where | Same as characteristic polynomial | 8 | 2 | 16 | , | |||

Diagonalizable over field:F3 with distinct diagonal entries |
Same as characteristic polynomial | 12 | 1 | 12 | ||||||

Total | NA | NA | NA | NA | NA | 8 | 48 | NA |