Properties

Label 37T4
Order \(148\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{37}:C_{4}$

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

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

Group invariants

Order:  $148=2^{2} \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [148, 3]
Character table:   
      2  2  2  2  2   .   .   .   .   .   .   .   .   .
     37  1  .  .  .   1   1   1   1   1   1   1   1   1

        1a 4a 4b 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i
     2P 1a 2a 2a 1a 37b 37d 37a 37f 37h 37i 37c 37g 37e
     3P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f
     5P 1a 4a 4b 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a
     7P 1a 4b 4a 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a
    11P 1a 4b 4a 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g
    13P 1a 4a 4b 2a 37d 37f 37b 37i 37g 37e 37a 37c 37h
    17P 1a 4a 4b 2a 37g 37c 37h 37a 37f 37b 37e 37i 37d
    19P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f
    23P 1a 4b 4a 2a 37h 37g 37e 37c 37d 37a 37i 37f 37b
    29P 1a 4a 4b 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g
    31P 1a 4b 4a 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i
    37P 1a 4a 4b 2a  1a  1a  1a  1a  1a  1a  1a  1a  1a

X.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
X.3      1  A -A -1   1   1   1   1   1   1   1   1   1
X.4      1 -A  A -1   1   1   1   1   1   1   1   1   1
X.5      4  .  .  .   B   F   D   E   I   C   H   J   G
X.6      4  .  .  .   C   G   E   I   D   J   F   B   H
X.7      4  .  .  .   D   B   H   F   G   E   J   I   C
X.8      4  .  .  .   E   C   F   G   H   I   B   D   J
X.9      4  .  .  .   F   E   B   C   J   G   D   H   I
X.10     4  .  .  .   G   I   C   J   B   H   E   F   D
X.11     4  .  .  .   H   D   J   B   C   F   I   G   E
X.12     4  .  .  .   I   J   G   H   F   D   C   E   B
X.13     4  .  .  .   J   H   I   D   E   B   G   C   F

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