Properties

Label 45T49
Order \(360\)
n \(45\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_6$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $49$
Group :  $A_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(4,44,15,27)(5,43,13,25)(6,45,14,26)(7,11,38,29)(8,10,39,30)(9,12,37,28)(16,24)(17,22,18,23)(19,35,40,32)(20,36,41,31)(21,34,42,33), (1,25,34,45,32)(2,26,36,44,33)(3,27,35,43,31)(4,16,23,42,38)(5,18,22,40,37)(6,17,24,41,39)(7,10,13,21,28)(8,12,14,19,29)(9,11,15,20,30)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: None

Degree 9: None

Degree 15: $A_6$ x 2

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, 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, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $45$ $2$ $( 4,15)( 5,13)( 6,14)( 7,38)( 8,39)( 9,37)(10,30)(11,29)(12,28)(17,18)(19,40) (20,41)(21,42)(22,23)(25,43)(26,45)(27,44)(31,36)(32,35)(33,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $ $90$ $4$ $( 2, 3)( 4,32,20,43)( 5,33,19,45)( 6,31,21,44)( 7,23,39,18)( 8,22,38,17) ( 9,24,37,16)(10,29,11,30)(12,28)(13,26,40,34)(14,27,42,36)(15,25,41,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,22,30)( 8,23,29)( 9,24,28)(10,39,18)(11,38,17) (12,37,16)(13,21,41)(14,20,40)(15,19,42)(25,44,33)(26,43,31)(27,45,32) (34,35,36)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $72$ $5$ $( 1, 4, 9,11,32)( 2, 5, 8,12,31)( 3, 6, 7,10,33)(13,36,42,23,44) (14,34,40,24,43)(15,35,41,22,45)(16,38,25,19,30)(17,37,26,21,28) (18,39,27,20,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 4,35)( 2, 5,36)( 3, 6,34)( 7,25,14)( 8,27,13)( 9,26,15)(10,16,38) (11,18,37)(12,17,39)(19,24,43)(20,22,44)(21,23,45)(28,31,42)(29,32,41) (30,33,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ $72$ $5$ $( 1, 4,14,42,20)( 2, 6,13,40,21)( 3, 5,15,41,19)( 7,39,44,28,31) ( 8,37,43,29,33)( 9,38,45,30,32)(10,16,26,36,23)(11,18,27,34,24) (12,17,25,35,22)$

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

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

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