Properties

Label 36T555
Order \(360\)
n \(36\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_6$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $555$
Group :  $A_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,36)(2,16,33)(3,17,34)(4,15,31)(5,18,32)(6,14,35)(7,26,19)(8,29,23)(9,30,20)(10,28,21)(11,25,22)(12,27,24), (1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,34,22)(14,35,21)(15,36,24)(16,31,20)(17,32,19)(18,33,23)(25,30,29)(26,28,27)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $A_6$ x 2

Degree 9: None

Degree 12: None

Degree 18: None

Low degree siblings

6T15 x 2, 10T26

Siblings are shown with degree $\leq 10$

Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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

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

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

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