# Element structure of projective general linear group:PGL(2,9)

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## Contents |

This article gives specific information, namely, element structure, about a particular group, namely: projective general linear group:PGL(2,9).

View element structure of particular groups | View other specific information about projective general linear group:PGL(2,9)

This article describes the element structure of projective general linear group:PGL(2,9), which is the projective general linear group of degree two over field:F9.

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

Note that this group is *not* isomorphic to symmetric group:S6, even though it has a subgroup of index two that is isomorphic to alternating group:A6.

## Conjugacy class structure

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

There is a total of conjugacy classes.

Nature of conjugacy class upstairs in | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements |
---|---|---|---|---|---|---|

Diagonalizable over field:F9 with equal diagonal entries, hence a scalar | where | 1 | 1 | 1 | ||

Diagonalizable over , not over , eigenvalues are negatives of each other. | Pair of mutually negative conjugate elements of . All such pairs identified. | , a nonzero non-square | Same as characteristic polynomial | 36 | 1 | |

Diagonalizable over with mutually negative diagonal entries. | , all such pairs identified. | , all identified | Same as characteristic polynomial | 45 | 1 | 45 |

Diagonalizable over , not over , eigenvalues are not negatives of each other. | Pair of conjugate elements of . Each pair identified with anything obtained by multiplying both elements of it by an element of . | , , irreducible; with identification. | Same as characteristic polynomial | 72 | 4 | 288 |

Not diagonal, has Jordan block of size two | (multiplicity 2). Each conjugacy class has one representative of each type. | Same as characteristic polynomial | 80 | 1 | 80 | |

Diagonalizable over with distinct diagonal entries whose sum is not zero. |
where and . The pairs and are identified. | , again with identification. | Same as characteristic polynomial. | 90 | 3 | 270 |

Total | NA | NA | NA | NA | 11 | 720 |