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

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   5  2   2  6  4  1  3   3  3   .   .  .  .  .   2  1  .
      3   4   1  1   .  1  .  3  1   3  1   1   1  4  2  2   1  1  .
      5   1   1  1   1  .  .  .  .   1  .   1   1  .  .  .   .  .  .
      7   1   .  .   .  .  .  .  .   .  .   .   .  .  .  .   .  .  1

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

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

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