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Properties

Label 12T183
Order \(720\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $183$
CHM label :  $S_{6}(12)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5)(2,4,12), (2,4,6,8,10)(3,5,7,9,11), (1,12)(2,3)(4,5)(6,7)(8,11)(9,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_6$

Low degree siblings

6T16 x 2, 10T32, 12T183, 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, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 6, 8,10)( 7, 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $45$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 5, 5, 1, 1 $ $144$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 6, 2, 2, 2 $ $120$ $6$ $( 1, 2)( 3,12)( 4, 5)( 6, 9,10, 7, 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 7)( 5, 6)( 8,11)( 9,10)$
$ 4, 4, 2, 2 $ $90$ $4$ $( 1, 2, 5, 6)( 3, 4, 7,12)( 8, 9)(10,11)$
$ 6, 6 $ $120$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 4, 4, 2, 2 $ $90$ $4$ $( 1, 3, 5, 7)( 2, 4, 6,12)( 8,10)( 9,11)$
$ 3, 3, 3, 3 $ $40$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$

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  1  4  .  4  1  4  3  1  3  1
      3  2  2  .  .  1  1  1  .  1  .  2
      5  1  .  .  1  .  .  .  .  .  .  .

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

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