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Properties

Label 6T15
Order \(360\)
n \(6\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_6$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $15$
Group :  $A_6$
CHM label :  $A6$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3), (1,2)(3,4,5,6)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 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 $ $1$ $1$ $()$
$ 3, 1, 1, 1 $ $40$ $3$ $(4,5,6)$
$ 2, 2, 1, 1 $ $45$ $2$ $(3,4)(5,6)$
$ 5, 1 $ $72$ $5$ $(2,3,4,5,6)$
$ 5, 1 $ $72$ $5$ $(2,3,4,6,5)$
$ 4, 2 $ $90$ $4$ $(1,2)(3,4,5,6)$
$ 3, 3 $ $40$ $3$ $(1,2,3)(4,5,6)$

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