Properties

Label 12T76
Order \(120\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2\times A_5$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $76$
Group :  $C_2\times A_5$
CHM label :  $[2]L(6)_{6}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10)(3,5,7,9,11), (4,10)(5,11)(6,8)(7,9), (1,2)(3,12)(4,11)(5,10), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $2$

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: None

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Low degree siblings

10T11, 12T75, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 5, 5, 1, 1 $ $12$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$
$ 5, 5, 1, 1 $ $12$ $5$ $( 2, 6,10, 4, 8)( 3, 7,11, 5, 9)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$
$ 6, 6 $ $20$ $6$ $( 1, 2, 5,12, 3, 4)( 6,11, 9, 7,10, 8)$
$ 10, 2 $ $12$ $10$ $( 1, 2, 6, 5, 9,12, 3, 7, 4, 8)(10,11)$
$ 3, 3, 3, 3 $ $20$ $3$ $( 1, 2, 6)( 3, 7,12)( 4,11, 9)( 5,10, 8)$
$ 10, 2 $ $12$ $10$ $( 1, 3, 5, 8,10,12, 2, 4, 9,11)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

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

        1a 2a 5a 5b 2b 6a 10a 3a 10b 2c
     2P 1a 1a 5b 5a 1a 3a  5b 3a  5a 1a
     3P 1a 2a 5b 5a 2b 2c 10b 1a 10a 2c
     5P 1a 2a 1a 1a 2b 6a  2c 3a  2c 2c
     7P 1a 2a 5b 5a 2b 6a 10b 3a 10a 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  A *A -1  .   A  .  *A  3
X.4      3 -1 *A  A -1  .  *A  .   A  3
X.5      3  1  A *A -1  .  -A  . -*A -3
X.6      3  1 *A  A -1  . -*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