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Properties

Label 10T32
Order \(720\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $S_{6}$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $32$
Group :  $S_{6}$
CHM label :  $S_{6}(10)=L(10):2$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7), (3,6)(4,7)(5,8)
$|\Aut(F/K)|$:  $1$

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 5: None

Low degree siblings

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

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

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

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