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Properties

Label 9T33
Degree $9$
Order $181440$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_9$

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Show commands: Magma

magma: G := TransitiveGroup(9, 33);
 

Group action invariants

Degree $n$:  $9$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_9$
CHM label:  $A9$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3), (3,4,5,6,7,8,9)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $181440=2^{6} \cdot 3^{4} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  181440.b
magma: IdentifyGroup(G);
 
Character table:    
      2   6  2   3   .   .  .  .  1  6  4  .   5  3  3   2  1  .   2
      3   4  1   3   1   1  4  2  3  1  .  2   1  1  1   1  1  .   .
      5   1  1   1   1   1  .  .  .  .  .  .   1  .  .   .  .  .   1
      7   1  .   .   .   .  .  .  .  .  .  .   .  .  .   .  .  1   .

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

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

magma: CharacterTable(G);