Properties

Label 40T304
Order \(360\)
n \(40\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_6$

Learn more about

Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $304$
Group :  $A_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,34,7,27)(2,21,33,8,28)(3,23,36,5,26)(4,24,35,6,25)(9,32,38,19,15)(10,31,37,20,16)(11,30,39,18,13)(12,29,40,17,14), (1,3,2,4)(5,14,10,29)(6,13,9,30)(7,16,11,31)(8,15,12,32)(17,40,34,27)(18,39,33,28)(19,38,36,26)(20,37,35,25)(21,23,22,24)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: $\PSL(2,9)$

Degree 20: $A_6$

Low degree siblings

6T15 x 2, 10T26

Siblings are shown with degree $\leq 10$

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 5, 9,21)( 6,10,22)( 7,12,23)( 8,11,24)(13,33,25)(14,34,26)(15,36,28) (16,35,27)(17,29,38)(18,30,37)(19,32,39)(20,31,40)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $40$ $3$ $( 5,15,20)( 6,16,19)( 7,14,18)( 8,13,17)( 9,36,31)(10,35,32)(11,33,29) (12,34,30)(21,28,40)(22,27,39)(23,26,37)(24,25,38)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $45$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,22)(10,21)(11,23)(12,24)(13,18)(14,17)(15,19) (16,20)(25,30)(26,29)(27,31)(28,32)(33,37)(34,38)(35,40)(36,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $90$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,34,22,38)(10,33,21,37)(11,36,23,39)(12,35,24,40) (13,31,18,27)(14,32,17,28)(15,30,19,25)(16,29,20,26)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $72$ $5$ $( 1, 5,13,33,31)( 2, 6,14,34,32)( 3, 8,16,35,29)( 4, 7,15,36,30) ( 9,26,20,22,39)(10,25,19,21,40)(11,28,17,23,37)(12,27,18,24,38)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $72$ $5$ $( 1, 5,26,34,20)( 2, 6,25,33,19)( 3, 8,28,36,17)( 4, 7,27,35,18) ( 9,40,22,14,32)(10,39,21,13,31)(11,38,23,15,30)(12,37,24,16,29)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 118]
Character table:   
     2  3  .  .  3  2  .  .
     3  2  2  2  .  .  .  .
     5  1  .  .  .  .  1  1

       1a 3a 3b 2a 4a 5a 5b
    2P 1a 3a 3b 1a 2a 5b 5a
    3P 1a 1a 1a 2a 4a 5b 5a
    5P 1a 3a 3b 2a 4a 1a 1a

X.1     1  1  1  1  1  1  1
X.2     5  2 -1  1 -1  .  .
X.3     5 -1  2  1 -1  .  .
X.4     8 -1 -1  .  .  A *A
X.5     8 -1 -1  .  . *A  A
X.6     9  .  .  1  1 -1 -1
X.7    10  1  1 -2  .  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5