Properties

Label 37T7
Order \(444\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{12}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$

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

Group invariants

Order:  $444=2^{2} \cdot 3 \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [444, 7]
Character table:   
      2  2  2   2   2   2   2   2   2   2   2  2  2   .   .   .
      3  1  1   1   1   1   1   1   1   1   1  1  1   .   .   .
     37  1  .   .   .   .   .   .   .   .   .  .  .   1   1   1

        1a 4a 12a  3a  6a 12b 12c  3b  6b 12d 4b 2a 37a 37b 37c
     2P 1a 2a  6b  3b  3a  6a  6a  3a  3b  6b 2a 1a 37b 37c 37a
     3P 1a 4b  4b  1a  2a  4a  4b  1a  2a  4a 4a 2a 37c 37a 37b
     5P 1a 4a 12c  3b  6b 12d 12a  3a  6a 12b 4b 2a 37c 37a 37b
     7P 1a 4b 12d  3a  6a 12c 12b  3b  6b 12a 4a 2a 37c 37a 37b
    11P 1a 4b 12b  3b  6b 12a 12d  3a  6a 12c 4a 2a 37a 37b 37c
    13P 1a 4a 12a  3a  6a 12b 12c  3b  6b 12d 4b 2a 37c 37a 37b
    17P 1a 4a 12c  3b  6b 12d 12a  3a  6a 12b 4b 2a 37b 37c 37a
    19P 1a 4b 12d  3a  6a 12c 12b  3b  6b 12a 4a 2a 37c 37a 37b
    23P 1a 4b 12b  3b  6b 12a 12d  3a  6a 12c 4a 2a 37a 37b 37c
    29P 1a 4a 12c  3b  6b 12d 12a  3a  6a 12b 4b 2a 37a 37b 37c
    31P 1a 4b 12d  3a  6a 12c 12b  3b  6b 12a 4a 2a 37a 37b 37c
    37P 1a 4a 12a  3a  6a 12b 12c  3b  6b 12d 4b 2a  1a  1a  1a

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

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