Properties

Label 37T6
Order \(333\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{9}$

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Group action invariants

Degree $n$ :  $37$
Transitive number $t$ :  $6$
Group :  $C_{37}:C_{9}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (1,16,34,26,9,33,10,12,7)(2,32,31,15,18,29,20,24,14)(3,11,28,4,27,25,30,36,21)(5,6,22,19,8,17,13,23,35)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2, 8,13,11,34,10,27,35,17)( 3,15,25,21,30,19,16,32,33)( 4,22,37,31,26,28, 5, 29,12)( 6,36,24,14,18, 9,20,23, 7)$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2,10, 8,27,13,35,11,17,34)( 3,19,15,16,25,32,21,33,30)( 4,28,22, 5,37,29,31, 12,26)( 6, 9,36,20,24,23,14, 7,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $37$ $3$ $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2,13,34,27,17, 8,11,10,35)( 3,25,30,16,33,15,21,19,32)( 4,37,26, 5,12,22,31, 28,29)( 6,24,18,20, 7,36,14, 9,23)$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2,17,35,27,10,34,11,13, 8)( 3,33,32,16,19,30,21,25,15)( 4,12,29, 5,28,26,31, 37,22)( 6, 7,23,20, 9,18,14,24,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $37$ $3$ $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2,34,17,11,35,13,27, 8,10)( 3,30,33,21,32,25,16,15,19)( 4,26,12,31,29,37, 5, 22,28)( 6,18, 7,14,23,24,20,36, 9)$
$ 9, 9, 9, 9, 1 $ $37$ $9$ $( 2,35,10,11, 8,17,27,34,13)( 3,32,19,21,15,33,16,30,25)( 4,29,28,31,22,12, 5, 26,37)( 6,23, 9,14,36, 7,20,18,24)$
$ 37 $ $9$ $37$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$
$ 37 $ $9$ $37$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$
$ 37 $ $9$ $37$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$
$ 37 $ $9$ $37$ $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$

Group invariants

Order:  $333=3^{2} \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [333, 3]
Character table:   
      3  2  2  2  2  2  2  2  2  2   .   .   .   .
     37  1  .  .  .  .  .  .  .  .   1   1   1   1

        1a 9a 9b 3a 9c 9d 3b 9e 9f 37a 37b 37c 37d
     2P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37b 37c 37d 37a
     3P 1a 3a 3b 1a 3b 3b 1a 3a 3a 37c 37d 37a 37b
     5P 1a 9b 9f 3b 9a 9e 3a 9c 9d 37d 37a 37b 37c
     7P 1a 9f 9d 3a 9b 9c 3b 9a 9e 37a 37b 37c 37d
    11P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37c 37d 37a 37b
    13P 1a 9e 9c 3a 9d 9b 3b 9f 9a 37d 37a 37b 37c
    17P 1a 9d 9e 3b 9f 9a 3a 9b 9c 37d 37a 37b 37c
    19P 1a 9a 9b 3a 9c 9d 3b 9e 9f 37d 37a 37b 37c
    23P 1a 9b 9f 3b 9a 9e 3a 9c 9d 37d 37a 37b 37c
    29P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37b 37c 37d 37a
    31P 1a 9e 9c 3a 9d 9b 3b 9f 9a 37b 37c 37d 37a
    37P 1a 9a 9b 3a 9c 9d 3b 9e 9f  1a  1a  1a  1a

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(9)^2-E(9)^5
C = E(9)^7
D = E(9)^5
E = E(37)+E(37)^7+E(37)^9+E(37)^10+E(37)^12+E(37)^16+E(37)^26+E(37)^33+E(37)^34
F = E(37)^2+E(37)^14+E(37)^15+E(37)^18+E(37)^20+E(37)^24+E(37)^29+E(37)^31+E(37)^32