Properties

Label 30T88
Order \(360\)
n \(30\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_6$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $88$
Group :  $A_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,6)(7,13)(8,14)(9,17)(10,18)(11,12)(15,20)(16,19)(21,24)(22,23)(25,28)(26,27)(29,30), (1,19,11,7,3)(2,20,12,8,4)(5,29,23,15,9)(6,30,24,16,10)(13,25,17,27,21)(14,26,18,28,22)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 5: None

Degree 6: $A_6$

Degree 10: None

Degree 15: $A_6$

Low degree siblings

6T15 x 2, 10T26, 15T20 x 2, 20T89, 30T88, 36T555, 40T304, 45T49

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

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

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

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