Properties

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

Related objects

Learn more about

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

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  4   5  2   3  3  3   2   .   .  .  .  1  1  .  .   2
      3   4  1  .   1  1   3  1  1   1   1   1  4  2  3  1  2  .   .
      5   1  .  .   1  1   1  .  .   .   1   1  .  .  .  .  .  .   1
      7   1  .  .   .  .   .  .  .   .   .   .  .  .  .  .  .  1   .

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

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

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