Properties

Label 30T38
Order \(150\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:S_3$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $38$
Group :  $C_5^2:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,22)(2,11,21)(3,13,24)(4,14,23)(5,15,26)(6,16,25)(7,18,27)(8,17,28)(9,20,30)(10,19,29), (1,27,8,3,14,10,20,15,25,21)(2,28,7,4,13,9,19,16,26,22)(5,30,24,17,11,6,29,23,18,12)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: None

Degree 15: $(C_5^2 : C_3):C_2$

Low degree siblings

15T13, 15T14, 25T16, 30T37

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3,10,15,21,27)( 4, 9,16,22,28)( 5,29,24,18,11)( 6,30,23,17,12)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3,15,27,10,21)( 4,16,28, 9,22)( 5,24,11,29,18)( 6,23,12,30,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,23) (22,24)(25,26)(27,30)(28,29)$
$ 10, 10, 10 $ $15$ $10$ $( 1, 3, 8,10,14,15,20,21,25,27)( 2, 4, 7, 9,13,16,19,22,26,28)( 5,30,24,17,11, 6,29,23,18,12)$
$ 10, 10, 10 $ $15$ $10$ $( 1, 3,14,15,25,27, 8,10,20,21)( 2, 4,13,16,26,28, 7, 9,19,22)( 5,23,11,30,18, 6,24,12,29,17)$
$ 10, 10, 10 $ $15$ $10$ $( 1, 3,20,21, 8,10,25,27,14,15)( 2, 4,19,22, 7, 9,26,28,13,16)( 5,17,29,12,24, 6,18,30,11,23)$
$ 10, 10, 10 $ $15$ $10$ $( 1, 3,25,27,20,21,14,15, 8,10)( 2, 4,26,28,19,22,13,16, 7, 9)( 5,12,18,23,29, 6,11,17,24,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)(19,21,24) (20,22,23)(25,28,30)(26,27,29)$
$ 5, 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,10,15,21,27)( 4, 9,16,22,28) ( 5,24,11,29,18)( 6,23,12,30,17)$
$ 5, 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,15,27,10,21)( 4,16,28, 9,22) ( 5,18,29,11,24)( 6,17,30,12,23)$
$ 5, 5, 5, 5, 5, 5 $ $3$ $5$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,27,21,15,10)( 4,28,22,16, 9) ( 5,29,24,18,11)( 6,30,23,17,12)$
$ 5, 5, 5, 5, 5, 5 $ $3$ $5$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,21,10,27,15)( 4,22, 9,28,16) ( 5,29,24,18,11)( 6,30,23,17,12)$

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [150, 5]
Character table:   
      2  1  .  .  1   1   1   1   1  .  1  1  1  1
      3  1  .  .  .   .   .   .   .  1  .  .  .  .
      5  2  2  2  1   1   1   1   1  .  2  2  2  2

        1a 5a 5b 2a 10a 10b 10c 10d 3a 5c 5d 5e 5f
     2P 1a 5b 5a 1a  5c  5d  5f  5e 3a 5d 5e 5f 5c
     3P 1a 5b 5a 2a 10c 10a 10d 10b 1a 5f 5c 5d 5e
     5P 1a 1a 1a 2a  2a  2a  2a  2a 3a 1a 1a 1a 1a
     7P 1a 5b 5a 2a 10b 10d 10a 10c 3a 5d 5e 5f 5c

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

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