Properties

Label 6T14
Degree $6$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PGL(2,5)$

Related objects

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Group action invariants

Degree $n$:  $6$
Transitive number $t$:  $14$
Group:  $\PGL(2,5)$
CHM label:  $L(6):2 = PGL(2,5) = S_{5}(6)$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2)(3,4)(5,6), (1,2,3,4,6)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

5T5, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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 $ $1$ $1$ $()$
$ 4, 1, 1 $ $30$ $4$ $(3,4,6,5)$
$ 2, 2, 1, 1 $ $15$ $2$ $(3,6)(4,5)$
$ 5, 1 $ $24$ $5$ $(2,3,5,4,6)$
$ 2, 2, 2 $ $10$ $2$ $(1,2)(3,4)(5,6)$
$ 3, 3 $ $20$ $3$ $(1,2,3)(4,5,6)$
$ 6 $ $20$ $6$ $(1,2,3,6,5,4)$

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [120, 34]
Character table:   
     2  3  2  3  .  2  1  1
     3  1  .  .  .  1  1  1
     5  1  .  .  1  .  .  .

       1a 4a 2a 5a 2b 3a 6a
    2P 1a 2a 1a 5a 1a 3a 3a
    3P 1a 4a 2a 5a 2b 1a 2b
    5P 1a 4a 2a 1a 2b 3a 6a

X.1     1  1  1  1  1  1  1
X.2     1 -1  1  1 -1  1 -1
X.3     4  .  . -1 -2  1  1
X.4     4  .  . -1  2  1 -1
X.5     5 -1  1  .  1 -1  1
X.6     5  1  1  . -1 -1 -1
X.7     6  . -2  1  .  .  .