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Properties

Label 10T11
Order \(120\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $11$
Group :  $A_5\times C_2$
CHM label :  $A(5)[x]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,10)(5,7,9), (1,6)(2,7)(3,8)(4,9)(5,10), (1,3,5,7,9)(2,4,6,8,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
60:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $A_5$

Low degree siblings

12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1 $ $20$ $3$ $( 3, 5, 9)( 4, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 4)( 3, 5)( 7, 9)( 8,10)$
$ 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$
$ 10 $ $12$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2, 3, 6, 7, 8)( 4, 9)( 5,10)$
$ 10 $ $12$ $10$ $( 1, 2, 3,10, 9, 6, 7, 8, 5, 4)$
$ 5, 5 $ $12$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $12$ $5$ $( 1, 3, 5, 9, 7)( 2, 6, 8,10, 4)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [120, 35]
Character table:    
      2  3  1  3  3   1  1   1  1  1  3
      3  1  1  .  .   .  1   .  .  .  1
      5  1  .  .  .   1  .   1  1  1  1

        1a 3a 2a 2b 10a 6a 10b 5a 5b 2c
     2P 1a 3a 1a 1a  5a 3a  5b 5b 5a 1a
     3P 1a 1a 2a 2b 10b 2c 10a 5b 5a 2c
     5P 1a 3a 2a 2b  2c 6a  2c 1a 1a 2c
     7P 1a 3a 2a 2b 10b 6a 10a 5b 5a 2c

X.1      1  1  1  1   1  1   1  1  1  1
X.2      1  1  1 -1  -1 -1  -1  1  1 -1
X.3      3  . -1 -1   A  .  *A *A  A  3
X.4      3  . -1 -1  *A  .   A  A *A  3
X.5      3  . -1  1 -*A  .  -A  A *A -3
X.6      3  . -1  1  -A  . -*A *A  A -3
X.7      4  1  .  .  -1  1  -1 -1 -1  4
X.8      4  1  .  .   1 -1   1 -1 -1 -4
X.9      5 -1  1  1   . -1   .  .  .  5
X.10     5 -1  1 -1   .  1   .  .  . -5

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5