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Properties

Label 30T20
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times A_4$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $20$
Group :  $D_5\times A_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,5,8,10)(2,3,6,7,9)(11,14,16,17,19,12,13,15,18,20)(21,24,26,27,30,22,23,25,28,29), (1,13,27,2,14,28)(3,12,30,10,16,25)(4,11,29,9,15,26)(5,19,22,7,17,23)(6,20,21,8,18,24)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$
12:  $A_4$
24:  $A_4\times C_2$
30:  $D_5\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 6: $A_4\times C_2$

Degree 10: None

Degree 15: $D_5\times C_3$

Low degree siblings

20T37, 30T28, 40T65

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 39]
Character table:    
      2  3  3  3  3   2  2   2  2   .   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   1   .   1  1

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

X.1      1  1  1  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   1  -1   1  1
X.3      1  1 -1 -1   1  1   1  1   C  -C  C   C  /C -/C  /C /C
X.4      1  1 -1 -1   1  1   1  1  /C -/C /C  /C   C  -C   C  C
X.5      1  1  1  1   1  1   1  1   C   C  C   C  /C  /C  /C /C
X.6      1  1  1  1   1  1   1  1  /C  /C /C  /C   C   C   C  C
X.7      2  2  .  .   A  A  *A *A   A   .  2  *A  *A   .   A  2
X.8      2  2  .  .  *A *A   A  A  *A   .  2   A   A   .  *A  2
X.9      2  2  .  .   A  A  *A *A   D   .  F   E  /E   .  /D /F
X.10     2  2  .  .   A  A  *A *A  /D   . /F  /E   E   .   D  F
X.11     2  2  .  .  *A *A   A  A   E   .  F   D  /D   .  /E /F
X.12     2  2  .  .  *A *A   A  A  /E   . /F  /D   D   .   E  F
X.13     3 -1 -1  3  -1  3  -1  3   .   .  .   .   .   .   .  .
X.14     3 -1  1 -3  -1  3  -1  3   .   .  .   .   .   .   .  .
X.15     6 -2  .  .  -A  B -*A *B   .   .  .   .   .   .   .  .
X.16     6 -2  .  . -*A *B  -A  B   .   .  .   .   .   .   .  .

A = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5
B = 3*E(5)+3*E(5)^4
  = (-3+3*Sqrt(5))/2 = 3b5
C = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
D = E(15)^7+E(15)^13
E = E(15)+E(15)^4
F = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3