Properties

Label 9T33
Order \(181440\)
n \(9\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_9$

Related objects

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Group action invariants

Degree $n$ :  $9$
Transitive number $t$ :  $33$
Group :  $A_9$
CHM label :  $A9$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3), (3,4,5,6,7,8,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

36T23796

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3 $ $2240$ $3$ $(1,4,2)(3,7,6)(5,9,8)$
$ 9 $ $20160$ $9$ $(1,3,9,4,7,8,2,6,5)$
$ 9 $ $20160$ $9$ $(1,4,9,7,2,3,8,5,6)$
$ 2, 2, 1, 1, 1, 1, 1 $ $378$ $2$ $(1,9)(5,7)$
$ 4, 2, 1, 1, 1 $ $7560$ $4$ $(1,7,9,5)(3,6)$
$ 2, 2, 2, 2, 1 $ $945$ $2$ $(1,9)(2,4)(3,6)(5,7)$
$ 4, 4, 1 $ $11340$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 9)$
$ 3, 1, 1, 1, 1, 1, 1 $ $168$ $3$ $(3,8,4)$
$ 3, 2, 2, 1, 1 $ $7560$ $6$ $(1,9)(3,4,8)(5,7)$
$ 4, 3, 2 $ $15120$ $12$ $(1,5,9,7)(2,6)(3,8,4)$
$ 3, 3, 1, 1, 1 $ $3360$ $3$ $(1,3,6)(4,9,5)$
$ 6, 2, 1 $ $30240$ $6$ $(1,5,3,4,6,9)(2,7)$
$ 5, 1, 1, 1, 1 $ $3024$ $5$ $(1,2,8,5,7)$
$ 5, 3, 1 $ $12096$ $15$ $(1,5,2,7,8)(3,6,9)$
$ 5, 3, 1 $ $12096$ $15$ $(1,5,2,7,8)(3,9,6)$
$ 5, 2, 2 $ $9072$ $10$ $(1,5)(2,4,6,3,8)(7,9)$
$ 7, 1, 1 $ $25920$ $7$ $(2,3,7,5,8,4,9)$

Group invariants

Order:  $181440=2^{6} \cdot 3^{4} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2   6  6  1  1  .   3  2   .   .  .  .  4   5  3  3   2   2  .
      3   4  1  3  1  .   3  1   1   1  4  2  .   1  1  1   1   .  2
      5   1  .  .  .  .   1  1   1   1  .  .  .   1  .  .   .   1  .
      7   1  .  .  .  1   .  .   .   .  .  .  .   .  .  .   .   .  .

         1a 2a 3a 6a 7a  3b 5a 15a 15b 3c 9a 4a  2b 4b 6b 12a 10a 9b
     2P  1a 1a 3a 3a 7a  3b 5a 15a 15b 3c 9a 2a  1a 2b 3b  6b  5a 9b
     3P  1a 2a 1a 2a 7a  1a 5a  5a  5a 1a 3c 4a  2b 4b 2b  4b 10a 3c
     5P  1a 2a 3a 6a 7a  3b 1a  3b  3b 3c 9a 4a  2b 4b 6b 12a  2b 9b
     7P  1a 2a 3a 6a 1a  3b 5a 15b 15a 3c 9a 4a  2b 4b 6b 12a 10a 9b
    11P  1a 2a 3a 6a 7a  3b 5a 15b 15a 3c 9a 4a  2b 4b 6b 12a 10a 9b
    13P  1a 2a 3a 6a 7a  3b 5a 15b 15a 3c 9a 4a  2b 4b 6b 12a 10a 9b

X.1       1  1  1  1  1   1  1   1   1  1  1  1   1  1  1   1   1  1
X.2       8  .  2  .  1   5  3   .   . -1 -1  .   4  2  1  -1  -1 -1
X.3      21 -3  .  .  .  -3  1   A  /A  3  .  1   1 -1  1  -1   1  .
X.4      21 -3  .  .  .  -3  1  /A   A  3  .  1   1 -1  1  -1   1  .
X.5      27  3  .  . -1   9  2  -1  -1  .  . -1   7  1  1   1   2  .
X.6      28 -4  1 -1  .  10  3   .   .  1  1  .   4  . -2   .  -1  1
X.7      35  3  2  .  .   5  .   .   . -1 -1 -1  -5 -1  1  -1   .  2
X.8      35  3  2  .  .   5  .   .   . -1  2 -1  -5 -1  1  -1   . -1
X.9      42  2  3 -1  .   . -3   .   . -3  .  2   6  .  .   .   1  .
X.10     48  .  .  . -1   6 -2   1   1  3  .  .   8  .  2   .  -2  .
X.11     56  .  2  .  .  11  1   1   1  2 -1  .  -4 -2 -1   1   1 -1
X.12     84  4  3  1  .  -6 -1  -1  -1  3  .  .   4  . -2   .  -1  .
X.13    105  1 -3  1  .  15  .   .   . -3  .  1   5 -1 -1  -1   .  .
X.14    120  8 -3 -1  1   .  .   .   .  3  .  .   .  .  .   .   .  .
X.15    162 -6  .  .  1   . -3   .   .  .  . -2   6  .  .   .   1  .
X.16    168  .  .  .  . -15  3   .   . -3  .  .   4 -2  1   1  -1  .
X.17    189 -3  .  .  .   9 -1  -1  -1  .  .  1 -11  1  1   1  -1  .
X.18    216  .  .  . -1  -9  1   1   1  .  .  .  -4  2 -1  -1   1  .

A = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15