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Properties

Label 6T16
Degree $6$
Order $720$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_6$

Related objects

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Group action invariants

Degree $n$:  $6$
Transitive number $t$:  $16$
Group:  $S_6$
CHM label:  $S6$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2), (1,2,3,4,5,6)

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

6T16, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1 $ $15$ $2$ $(5,6)$
$ 3, 1, 1, 1 $ $40$ $3$ $(4,5,6)$
$ 2, 2, 1, 1 $ $45$ $2$ $(3,4)(5,6)$
$ 4, 1, 1 $ $90$ $4$ $(3,4,5,6)$
$ 3, 2, 1 $ $120$ $6$ $(2,3)(4,5,6)$
$ 5, 1 $ $144$ $5$ $(2,3,4,5,6)$
$ 2, 2, 2 $ $15$ $2$ $(1,2)(3,4)(5,6)$
$ 4, 2 $ $90$ $4$ $(1,2)(3,4,5,6)$
$ 3, 3 $ $40$ $3$ $(1,2,3)(4,5,6)$
$ 6 $ $120$ $6$ $(1,2,3,4,5,6)$

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [720, 763]
Character table:    
      2  4  4  1  4  3  1  .  4  3  1  1
      3  2  1  2  .  .  1  .  1  .  2  1
      5  1  .  .  .  .  .  1  .  .  .  .

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

X.1      1 -1  1  1 -1 -1  1 -1  1  1 -1
X.2      5 -3  2  1 -1  .  .  1 -1 -1  1
X.3      9 -3  .  1  1  . -1 -3  1  .  .
X.4      5 -1 -1  1  1 -1  .  3 -1  2  .
X.5     10 -2  1 -2  .  1  .  2  .  1 -1
X.6     16  . -2  .  .  .  1  .  . -2  .
X.7      5  1 -1  1 -1  1  . -3 -1  2  .
X.8     10  2  1 -2  . -1  . -2  .  1  1
X.9      9  3  .  1 -1  . -1  3  1  .  .
X.10     5  3  2  1  1  .  . -1 -1 -1 -1
X.11     1  1  1  1  1  1  1  1  1  1  1