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Properties

Label 30T29
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2\times A_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $29$
Group :  $C_2\times A_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,11,2,15,12)(3,25,30,4,26,29)(5,13,21,6,14,22)(7,27,9,8,28,10)(17,24,19,18,23,20), (1,19,28,18,5,2,20,27,17,6)(3,13,12,30,24,4,14,11,29,23)(7,22,15,10,25,8,21,16,9,26)
$|\Aut(F/K)|$:  $2$

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 5: $A_5$

Degree 6: None

Degree 10: None

Degree 15: $A_5$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 3,10)( 4, 9)( 5,26)( 6,25)( 7,19)( 8,20)(13,27)(14,28)(15,16)(17,30)(18,29) (21,24)(22,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 3,22)( 4,21)( 5, 8)( 6, 7)( 9,24)(10,23)(11,12)(13,18)(14,17)(15,16)(19,25) (20,26)(27,29)(28,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$
$ 10, 10, 10 $ $12$ $10$ $( 1, 3,14,27, 9, 2, 4,13,28,10)( 5,26,30,15,18, 6,25,29,16,17)( 7,22,24,20,11, 8,21,23,19,12)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 3,19,25,22)( 2, 4,20,26,21)( 5,11, 7,29,28)( 6,12, 8,30,27) ( 9,23,14,15,18)(10,24,13,16,17)$
$ 6, 6, 6, 6, 6 $ $20$ $6$ $( 1, 3,24, 2, 4,23)( 5,21,13, 6,22,14)( 7,25,12, 8,26,11)( 9,20,30,10,19,29) (15,17,27,16,18,28)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,14,28, 9)( 2, 3,13,27,10)( 5,25,30,16,18)( 6,26,29,15,17) ( 7,21,24,19,11)( 8,22,23,20,12)$
$ 10, 10, 10 $ $12$ $10$ $( 1, 4,19,26,22, 2, 3,20,25,21)( 5,12, 7,30,28, 6,11, 8,29,27)( 9,24,14,16,18, 10,23,13,15,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 4,24)( 2, 3,23)( 5,22,13)( 6,21,14)( 7,26,12)( 8,25,11)( 9,19,30) (10,20,29)(15,18,27)(16,17,28)$

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  3  3   1  1  1  1   1  1
      3  1  .  .  1   .  .  1  .   .  1
      5  1  .  .  1   1  1  .  1   1  .

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

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

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