Properties

Label 37T8
Order \(666\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{18}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$

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

Group invariants

Order:  $666=2 \cdot 3^{2} \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [666, 7]
Character table:   
      2  1   .   .  1  1  1  1  1  1  1  1   1   1   1   1  1   1   1   1   1
      3  2   .   .  2  2  2  2  2  2  2  2   2   2   2   2  2   2   2   2   2
     37  1   1   1  .  .  .  .  .  .  .  .   .   .   .   .  .   .   .   .   .

        1a 37a 37b 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c  6a 2a  6b 18d 18e 18f
     2P 1a 37b 37a 3b 3a 9d 9f 9e 9b 9a 9c  9a  9c  9b  3a 1a  3b  9e  9d  9f
     3P 1a 37a 37b 1a 1a 3a 3a 3a 3b 3b 3b  6a  6a  6a  2a 2a  2a  6b  6b  6b
     5P 1a 37b 37a 3b 3a 9e 9d 9f 9a 9c 9b 18d 18f 18e  6b 2a  6a 18b 18a 18c
     7P 1a 37a 37b 3a 3b 9c 9a 9b 9e 9f 9d 18b 18c 18a  6a 2a  6b 18f 18d 18e
    11P 1a 37a 37b 3b 3a 9d 9f 9e 9b 9a 9c 18e 18d 18f  6b 2a  6a 18a 18c 18b
    13P 1a 37b 37a 3a 3b 9b 9c 9a 9f 9d 9e 18c 18a 18b  6a 2a  6b 18e 18f 18d
    17P 1a 37b 37a 3b 3a 9f 9e 9d 9c 9b 9a 18f 18e 18d  6b 2a  6a 18c 18b 18a
    19P 1a 37b 37a 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c  6a 2a  6b 18d 18e 18f
    23P 1a 37b 37a 3b 3a 9e 9d 9f 9a 9c 9b 18d 18f 18e  6b 2a  6a 18b 18a 18c
    29P 1a 37b 37a 3b 3a 9d 9f 9e 9b 9a 9c 18e 18d 18f  6b 2a  6a 18a 18c 18b
    31P 1a 37b 37a 3a 3b 9b 9c 9a 9f 9d 9e 18c 18a 18b  6a 2a  6b 18e 18f 18d
    37P 1a  1a  1a 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c  6a 2a  6b 18d 18e 18f

X.1      1   1   1  1  1  1  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  -1  -1 -1  -1  -1  -1  -1
X.3      1   1   1  1  1  B  B  B /B /B /B -/B -/B -/B  -1 -1  -1  -B  -B  -B
X.4      1   1   1  1  1 /B /B /B  B  B  B  -B  -B  -B  -1 -1  -1 -/B -/B -/B
X.5      1   1   1  1  1  B  B  B /B /B /B  /B  /B  /B   1  1   1   B   B   B
X.6      1   1   1  1  1 /B /B /B  B  B  B   B   B   B   1  1   1  /B  /B  /B
X.7      1   1   1  B /B  C  D  E /E /D /C -/D -/C -/E -/B -1  -B  -E  -C  -D
X.8      1   1   1  B /B  D  E  C /C /E /D -/E -/D -/C -/B -1  -B  -C  -D  -E
X.9      1   1   1  B /B  E  C  D /D /C /E -/C -/E -/D -/B -1  -B  -D  -E  -C
X.10     1   1   1 /B  B /C /D /E  E  D  C  -D  -C  -E  -B -1 -/B -/E -/C -/D
X.11     1   1   1 /B  B /E /C /D  D  C  E  -C  -E  -D  -B -1 -/B -/D -/E -/C
X.12     1   1   1 /B  B /D /E /C  C  E  D  -E  -D  -C  -B -1 -/B -/C -/D -/E
X.13     1   1   1  B /B  C  D  E /E /D /C  /D  /C  /E  /B  1   B   E   C   D
X.14     1   1   1  B /B  D  E  C /C /E /D  /E  /D  /C  /B  1   B   C   D   E
X.15     1   1   1  B /B  E  C  D /D /C /E  /C  /E  /D  /B  1   B   D   E   C
X.16     1   1   1 /B  B /C /D /E  E  D  C   D   C   E   B  1  /B  /E  /C  /D
X.17     1   1   1 /B  B /E /C /D  D  C  E   C   E   D   B  1  /B  /D  /E  /C
X.18     1   1   1 /B  B /D /E /C  C  E  D   E   D   C   B  1  /B  /C  /D  /E
X.19    18   A  *A  .  .  .  .  .  .  .  .   .   .   .   .  .   .   .   .   .
X.20    18  *A   A  .  .  .  .  .  .  .  .   .   .   .   .  .   .   .   .   .

A = E(37)+E(37)^3+E(37)^4+E(37)^7+E(37)^9+E(37)^10+E(37)^11+E(37)^12+E(37)^16+E(37)^21+E(37)^25+E(37)^26+E(37)^27+E(37)^28+E(37)^30+E(37)^33+E(37)^34+E(37)^36
  = (-1+Sqrt(37))/2 = b37
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = -E(9)^2-E(9)^5
D = E(9)^5
E = E(9)^2