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

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

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

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