Properties

Label 37T3
Order \(111\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{3}$

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Group action invariants

Degree $n$ :  $37$
Transitive number $t$ :  $3$
Group :  $C_{37}:C_{3}$
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), (1,26,10)(2,15,20)(3,4,30)(5,19,13)(6,8,23)(7,34,33)(9,12,16)(11,27,36)(14,31,29)(17,35,22)(18,24,32)(21,28,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $37$ $3$ $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $37$ $3$ $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$
$ 37 $ $3$ $37$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$
$ 37 $ $3$ $37$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$
$ 37 $ $3$ $37$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$
$ 37 $ $3$ $37$ $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$
$ 37 $ $3$ $37$ $( 1, 7,13,19,25,31,37, 6,12,18,24,30,36, 5,11,17,23,29,35, 4,10,16,22,28,34, 3, 9,15,21,27,33, 2, 8,14,20,26,32)$
$ 37 $ $3$ $37$ $( 1, 8,15,22,29,36, 6,13,20,27,34, 4,11,18,25,32, 2, 9,16,23,30,37, 7,14,21, 28,35, 5,12,19,26,33, 3,10,17,24,31)$
$ 37 $ $3$ $37$ $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$
$ 37 $ $3$ $37$ $( 1,12,23,34, 8,19,30, 4,15,26,37,11,22,33, 7,18,29, 3,14,25,36,10,21,32, 6, 17,28, 2,13,24,35, 9,20,31, 5,16,27)$
$ 37 $ $3$ $37$ $( 1,15,29, 6,20,34,11,25, 2,16,30, 7,21,35,12,26, 3,17,31, 8,22,36,13,27, 4, 18,32, 9,23,37,14,28, 5,19,33,10,24)$
$ 37 $ $3$ $37$ $( 1,18,35,15,32,12,29, 9,26, 6,23, 3,20,37,17,34,14,31,11,28, 8,25, 5,22, 2, 19,36,16,33,13,30,10,27, 7,24, 4,21)$
$ 37 $ $3$ $37$ $( 1,19,37,18,36,17,35,16,34,15,33,14,32,13,31,12,30,11,29,10,28, 9,27, 8,26, 7,25, 6,24, 5,23, 4,22, 3,21, 2,20)$
$ 37 $ $3$ $37$ $( 1,22, 6,27,11,32,16,37,21, 5,26,10,31,15,36,20, 4,25, 9,30,14,35,19, 3,24, 8,29,13,34,18, 2,23, 7,28,12,33,17)$

Group invariants

Order:  $111=3 \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [111, 1]
Character table:   
      3  1  1  1   .   .   .   .   .   .   .   .   .   .   .   .
     37  1  .  .   1   1   1   1   1   1   1   1   1   1   1   1

        1a 3a 3b 37a 37b 37c 37d 37e 37f 37g 37h 37i 37j 37k 37l
     2P 1a 3b 3a 37b 37c 37e 37a 37g 37i 37k 37j 37l 37f 37h 37d
     3P 1a 1a 1a 37c 37e 37g 37b 37k 37l 37h 37f 37d 37i 37j 37a
     5P 1a 3b 3a 37d 37a 37b 37l 37c 37j 37e 37k 37f 37h 37g 37i
     7P 1a 3a 3b 37f 37i 37l 37j 37d 37g 37a 37c 37k 37e 37b 37h
    11P 1a 3b 3a 37h 37j 37f 37k 37i 37c 37l 37a 37e 37b 37d 37g
    13P 1a 3a 3b 37d 37a 37b 37l 37c 37j 37e 37k 37f 37h 37g 37i
    17P 1a 3b 3a 37j 37f 37i 37h 37l 37e 37d 37b 37g 37c 37a 37k
    19P 1a 3a 3b 37d 37a 37b 37l 37c 37j 37e 37k 37f 37h 37g 37i
    23P 1a 3b 3a 37e 37g 37k 37c 37h 37d 37j 37i 37a 37l 37f 37b
    29P 1a 3b 3a 37i 37l 37d 37f 37a 37k 37b 37e 37h 37g 37c 37j
    31P 1a 3a 3b 37i 37l 37d 37f 37a 37k 37b 37e 37h 37g 37c 37j
    37P 1a 3a 3b  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(37)^6+E(37)^8+E(37)^23
C = E(37)+E(37)^10+E(37)^26
D = E(37)^3+E(37)^4+E(37)^30
E = E(37)^21+E(37)^25+E(37)^28
F = E(37)^5+E(37)^13+E(37)^19
G = E(37)^17+E(37)^22+E(37)^35