Properties

Label 30T31
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:S_4$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $31$
Group :  $C_5:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,2,24)(3,22,4,21)(5,29,6,30)(7,28,8,27)(9,26,10,25)(13,19)(14,20)(15,18)(16,17), (1,11,24,5,16,27,9,19,21,3,13,25,7,17,30)(2,12,23,6,15,28,10,20,22,4,14,26,8,18,29)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
10:  $D_{5}$
24:  $S_4$
30:  $D_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $S_4$

Degree 10: None

Degree 15: $D_{15}$

Low degree siblings

20T33, 30T19, 40T63

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 38]
Character table:   
      2  3  3  2  2  2   2   2  2   .  .   .   .   .
      3  1  .  .  .  1   .   .  1   1  1   1   1   1
      5  1  1  .  .  1   1   1  1   1  1   1   1   1

        1a 2a 4a 2b 5a 10a 10b 5b 15a 3a 15b 15c 15d
     2P 1a 1a 2a 1a 5b  5b  5a 5a 15c 3a 15d 15b 15a
     3P 1a 2a 4a 2b 5b 10b 10a 5a  5b 1a  5b  5a  5a
     5P 1a 2a 4a 2b 1a  2a  2a 1a  3a 3a  3a  3a  3a
     7P 1a 2a 4a 2b 5b 10b 10a 5a 15d 3a 15c 15a 15b
    11P 1a 2a 4a 2b 5a 10a 10b 5b 15b 3a 15a 15d 15c
    13P 1a 2a 4a 2b 5b 10b 10a 5a 15c 3a 15d 15b 15a

X.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
X.3      2  2  .  .  2   2   2  2  -1 -1  -1  -1  -1
X.4      2  2  .  .  A   A  *A *A   A  2   A  *A  *A
X.5      2  2  .  . *A  *A   A  A  *A  2  *A   A   A
X.6      2  2  .  . *A  *A   A  A   C -1   D   E   F
X.7      2  2  .  . *A  *A   A  A   D -1   C   F   E
X.8      2  2  .  .  A   A  *A *A   E -1   F   D   C
X.9      2  2  .  .  A   A  *A *A   F -1   E   C   D
X.10     3 -1 -1  1  3  -1  -1  3   .  .   .   .   .
X.11     3 -1  1 -1  3  -1  -1  3   .  .   .   .   .
X.12     6 -2  .  .  B  -A -*A *B   .  .   .   .   .
X.13     6 -2  .  . *B -*A  -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(15)^4+E(15)^11
D = E(15)+E(15)^14
E = E(15)^7+E(15)^8
F = E(15)^2+E(15)^13