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Properties

Label 30T50
Degree $30$
Order $240$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{16}$

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Group action invariants

Degree $n$:  $30$
Transitive number $t$:  $50$
Group:  $F_{16}$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22), (1,18,21,25,15,3,12,10,19,7,6,30,27,13,23)(2,17,22,26,16,4,11,9,20,8,5,29,28,14,24)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: None

Degree 10: None

Degree 15: $C_{15}$

Low degree siblings

16T447, 20T67

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

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [240, 191]
Character table:    
      2  4  4  .  .  .   .  .   .   .   .   .   .   .  .  .   .
      3  1  .  1  1  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  1  1   1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(5)^4
C = E(5)^3
D = E(15)^11
E = E(15)^2
F = E(15)^8
G = E(15)^14