Properties

Label 30T10
Order \(60\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_5$

Learn more about

Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $10$
Group :  $S_3\times D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,11,8,21,18)(2,28,12,7,22,17)(3,6,13,16,24,25)(4,5,14,15,23,26)(9,30,19,10,29,20), (1,19,7,26,13)(2,20,8,25,14)(3,11,9,17,15,24,21,29,28,5)(4,12,10,18,16,23,22,30,27,6)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
10:  $D_{5}$
12:  $D_{6}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $D_{6}$

Degree 10: $D_5$

Degree 15: $D_5\times S_3$

Low degree siblings

15T7, 30T8, 30T13

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

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [60, 8]
Character table:   
      2  2  2  2  2   .   1  1   .   1  1  1  1
      3  1  .  1  .   1   .  1   1   .  1  1  1
      5  1  1  .  .   1   1  .   1   1  1  1  1

        1a 2a 2b 2c 15a 10a 6a 15b 10b 5a 3a 5b
     2P 1a 1a 1a 1a 15b  5a 3a 15a  5b 5b 3a 5a
     3P 1a 2a 2b 2c  5a 10b 2b  5b 10a 5b 1a 5a
     5P 1a 2a 2b 2c  3a  2a 6a  3a  2a 1a 3a 1a
     7P 1a 2a 2b 2c 15b 10b 6a 15a 10a 5b 3a 5a
    11P 1a 2a 2b 2c 15a 10a 6a 15b 10b 5a 3a 5b
    13P 1a 2a 2b 2c 15b 10b 6a 15a 10a 5b 3a 5a

X.1      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  1
X.3      1 -1  1 -1   1  -1  1   1  -1  1  1  1
X.4      1  1 -1 -1   1   1 -1   1   1  1  1  1
X.5      2  . -2  .  -1   .  1  -1   .  2 -1  2
X.6      2  .  2  .  -1   . -1  -1   .  2 -1  2
X.7      2 -2  .  .   A  -A  .  *A -*A *A  2  A
X.8      2 -2  .  .  *A -*A  .   A  -A  A  2 *A
X.9      2  2  .  .   A   A  .  *A  *A *A  2  A
X.10     2  2  .  .  *A  *A  .   A   A  A  2 *A
X.11     4  .  .  .  -A   .  . -*A   .  B -2 *B
X.12     4  .  .  . -*A   .  .  -A   . *B -2  B

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)+2*E(5)^4
  = -1+Sqrt(5) = 2b5