Properties

Label 30T48
Order \(180\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2:(C_5:C_4)$

Learn more about

Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $48$
Group :  $C_3^2:(C_5:C_4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,15,4,11,3,7,14,6,10,2,9,13,5,12)(16,24,29,19,27,17,22,30,20,25,18,23,28,21,26), (1,26,3,25)(2,27)(4,23,6,22)(5,24)(7,20,9,19)(8,21)(10,17,12,16)(11,18)(13,29,15,28)(14,30)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  20T2
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $C_3^2:C_4$

Degree 10: $D_5$

Degree 15: None

Low degree siblings

30T48, 45T26

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,19,22,25,28) (17,20,23,26,29)(18,21,24,27,30)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,20,24,25,29,18,19,23,27, 28,17,21,22,26,30)$
$ 10, 10, 5, 5 $ $18$ $10$ $( 1, 4, 7,10,13)( 2, 6, 8,12,14, 3, 5, 9,11,15)(16,19,22,25,28) (17,21,23,27,29,18,20,24,26,30)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,19,22,25,28)(17,20,23,26,29) (18,21,24,27,30)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,20,24,25,29,18,19,23,27,28, 17,21,22,26,30)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,21,23,25,30,17,19,24,26,28, 18,20,22,27,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,22,28,19,25) (17,23,29,20,26)(18,24,30,21,27)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,23,30,19,26,18,22,29,21, 25,17,24,28,20,27)$
$ 10, 10, 5, 5 $ $18$ $10$ $( 1, 7,13, 4,10)( 2, 9,14, 6,11, 3, 8,15, 5,12)(16,22,28,19,25) (17,24,29,21,26,18,23,30,20,27)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,22,28,19,25)(17,23,29,20,26) (18,24,30,21,27)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,23,30,19,26,18,22,29,21,25, 17,24,28,20,27)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,24,29,19,27,17,22,30,20,25, 18,23,28,21,26)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $45$ $4$ $( 1,16)( 2,17, 3,18)( 4,28)( 5,29, 6,30)( 7,25)( 8,26, 9,27)(10,22) (11,23,12,24)(13,19)(14,20,15,21)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $45$ $4$ $( 1,16)( 2,18, 3,17)( 4,28)( 5,30, 6,29)( 7,25)( 8,27, 9,26)(10,22) (11,24,12,23)(13,19)(14,21,15,20)$

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 24]
Character table:   
      2  2  .  2  .  1   .   1   .   .   .  1   .   1   .   .   .  2  2
      3  2  2  .  2  2   2   .   2   2   2  2   2   .   2   2   2  .  .
      5  1  1  1  1  1   1   1   1   1   1  1   1   1   1   1   1  .  .

        1a 3a 2a 3b 5a 15a 10a 15b 15c 15d 5b 15e 10b 15f 15g 15h 4a 4b
     2P 1a 3a 1a 3b 5b 15e  5b 15f 15g 15h 5a 15b  5a 15a 15d 15c 2a 2a
     3P 1a 1a 2a 1a 5b  5b 10b  5b  5b  5b 5a  5a 10a  5a  5a  5a 4b 4a
     5P 1a 3a 2a 3b 1a  3a  2a  3a  3b  3b 1a  3a  2a  3a  3b  3b 4a 4b
     7P 1a 3a 2a 3b 5b 15e 10b 15f 15g 15h 5a 15b 10a 15a 15d 15c 4b 4a
    11P 1a 3a 2a 3b 5a 15a 10a 15b 15c 15d 5b 15e 10b 15f 15g 15h 4b 4a
    13P 1a 3a 2a 3b 5b 15f 10b 15e 15h 15g 5a 15a 10a 15b 15c 15d 4a 4b

X.1      1  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 -1 -1
X.3      1  1 -1  1  1   1  -1   1   1   1  1   1  -1   1   1   1  E -E
X.4      1  1 -1  1  1   1  -1   1   1   1  1   1  -1   1   1   1 -E  E
X.5      2  2 -2  2  A   A  -A   A   A   A *A  *A -*A  *A  *A  *A  .  .
X.6      2  2 -2  2 *A  *A -*A  *A  *A  *A  A   A  -A   A   A   A  .  .
X.7      2  2  2  2  A   A   A   A   A   A *A  *A  *A  *A  *A  *A  .  .
X.8      2  2  2  2 *A  *A  *A  *A  *A  *A  A   A   A   A   A   A  .  .
X.9      4 -2  .  1  4  -2   .  -2   1   1  4  -2   .  -2   1   1  .  .
X.10     4  1  . -2  4   1   .   1  -2  -2  4   1   .   1  -2  -2  .  .
X.11     4 -2  .  1  B  -A   .  -A   C  /C *B -*A   . -*A  /D   D  .  .
X.12     4 -2  .  1  B  -A   .  -A  /C   C *B -*A   . -*A   D  /D  .  .
X.13     4 -2  .  1 *B -*A   . -*A   D  /D  B  -A   .  -A   C  /C  .  .
X.14     4 -2  .  1 *B -*A   . -*A  /D   D  B  -A   .  -A  /C   C  .  .
X.15     4  1  . -2  B   C   .  /C  -A  -A *B  /D   .   D -*A -*A  .  .
X.16     4  1  . -2  B  /C   .   C  -A  -A *B   D   .  /D -*A -*A  .  .
X.17     4  1  . -2 *B   D   .  /D -*A -*A  B   C   .  /C  -A  -A  .  .
X.18     4  1  . -2 *B  /D   .   D -*A -*A  B  /C   .   C  -A  -A  .  .

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
C = -E(5)^2+2*E(5)^3
D = -E(5)+2*E(5)^4
E = -E(4)
  = -Sqrt(-1) = -i