Properties

Label 30T4
Order \(30\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_5$

Learn more about

Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $4$
Group :  $C_3\times D_5$
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)(3,21,9,28,15)(4,22,10,27,16)(5,24,11,29,17)(6,23,12,30,18)
$|\Aut(F/K)|$:  $30$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 6: $C_6$

Degree 10: $D_5$

Degree 15: $D_5\times C_3$

Low degree siblings

15T3

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

Group invariants

Order:  $30=2 \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [30, 2]
Character table:   
      2  1  1   .   1   .   1  .  1  .   .  1   .
      3  1  1   1   1   1   1  1  1  1   1  1   1
      5  1  .   1   .   1   .  1  1  1   1  1   1

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

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   A  -A  /A -/A  1 /A  1   A  A  /A
X.4      1 -1  /A -/A   A  -A  1  A  1  /A /A   A
X.5      1  1   A   A  /A  /A  1 /A  1   A  A  /A
X.6      1  1  /A  /A   A   A  1  A  1  /A /A   A
X.7      2  .   B   .  *B   . *B  2  B  *B  2   B
X.8      2  .  *B   .   B   .  B  2 *B   B  2  *B
X.9      2  .   C   .  /D   . *B  E  B   D /E  /C
X.10     2  .  /C   .   D   . *B /E  B  /D  E   C
X.11     2  .   D   .  /C   .  B  E *B   C /E  /D
X.12     2  .  /D   .   C   .  B /E *B  /C  E   D

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
C = E(15)^11+E(15)^14
D = E(15)^2+E(15)^8
E = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3