Properties

Label 37T5
Order \(222\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{6}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

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

Group invariants

Order:  $222=2 \cdot 3 \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [222, 1]
Character table:   
      2  1  1   1  1   1  1   .   .   .   .   .   .
      3  1  1   1  1   1  1   .   .   .   .   .   .
     37  1  .   .  .   .  .   1   1   1   1   1   1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(37)^3+E(37)^4+E(37)^7+E(37)^30+E(37)^33+E(37)^34
C = E(37)^2+E(37)^15+E(37)^17+E(37)^20+E(37)^22+E(37)^35
D = E(37)+E(37)^10+E(37)^11+E(37)^26+E(37)^27+E(37)^36
E = E(37)^5+E(37)^13+E(37)^18+E(37)^19+E(37)^24+E(37)^32
F = E(37)^9+E(37)^12+E(37)^16+E(37)^21+E(37)^25+E(37)^28
G = E(37)^6+E(37)^8+E(37)^14+E(37)^23+E(37)^29+E(37)^31