Properties

Label 37T2
Order \(74\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_{37}$

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Group action invariants

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

Low degree resolvents

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

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

Group invariants

Order:  $74=2 \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [74, 1]
Character table:   
      2  1  1   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .
     37  1  .   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

        1a 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i 37j 37k 37l 37m 37n 37o 37p
     2P 1a 1a 37b 37d 37f 37h 37j 37l 37n 37p 37r 37q 37o 37m 37k 37i 37g 37e
     3P 1a 2a 37c 37f 37i 37l 37o 37r 37p 37m 37j 37g 37d 37a 37b 37e 37h 37k
     5P 1a 2a 37e 37j 37o 37q 37l 37g 37b 37c 37h 37m 37r 37n 37i 37d 37a 37f
     7P 1a 2a 37g 37n 37p 37i 37b 37e 37l 37r 37k 37d 37c 37j 37q 37m 37f 37a
    11P 1a 2a 37k 37o 37d 37g 37r 37h 37c 37n 37l 37a 37j 37p 37e 37f 37q 37i
    13P 1a 2a 37m 37k 37b 37o 37i 37d 37q 37g 37f 37r 37e 37h 37p 37c 37j 37n
    17P 1a 2a 37q 37c 37n 37f 37k 37i 37h 37l 37e 37o 37b 37r 37a 37p 37d 37m
    19P 1a 2a 37r 37a 37q 37b 37p 37c 37o 37d 37n 37e 37m 37f 37l 37g 37k 37h
    23P 1a 2a 37n 37i 37e 37r 37d 37j 37m 37a 37o 37h 37f 37q 37c 37k 37l 37b
    29P 1a 2a 37h 37p 37m 37e 37c 37k 37r 37j 37b 37f 37n 37o 37g 37a 37i 37q
    31P 1a 2a 37f 37l 37r 37m 37g 37a 37e 37k 37q 37n 37h 37b 37d 37j 37p 37o
    37P 1a 2a  1a  1a  1a  1a  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   1   1   1
X.2      1 -1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
X.3      2  .   A   P   B   J   I   C   R   G   M   H   O   E   K   L   D   F
X.4      2  .   B   C   M   E   D   Q   F   K   H   R   J   A   P   I   G   O
X.5      2  .   C   E   Q   K   R   A   I   O   N   L   G   P   J   H   F   D
X.6      2  .   D   R   G   L   A   F   C   M   K   P   N   I   H   E   B   Q
X.7      2  .   E   K   A   O   L   P   H   D   B   M   F   J   G   N   I   R
X.8      2  .   F   I   O   H   C   D   A   N   J   E   M   R   L   P   Q   B
X.9      2  .   G   F   K   I   B   O   Q   H   P   C   L   D   R   A   M   N
X.10     2  .   H   N   R   B   K   L   J   C   F   O   A   M   Q   G   P   E
X.11     2  .   I   H   D   N   E   R   P   B   G   K   Q   L   M   J   A   C
X.12     2  .   J   G   E   F   N   K   M   I   A   B   R   O   D   Q   L   H
X.13     2  .   K   O   P   D   M   J   N   R   C   Q   I   G   F   B   H   L
X.14     2  .   L   M   I   Q   J   H   K   A   D   G   C   N   B   O   E   P
X.15     2  .   M   Q   H   A   G   N   O   P   R   F   E   B   C   D   K   J
X.16     2  .   N   B   L   C   O   M   G   E   I   D   P   Q   A   F   J   K
X.17     2  .   O   D   J   R   Q   G   B   L   E   A   H   F   I   C   N   M
X.18     2  .   P   J   C   G   H   E   L   F   Q   N   D   K   O   M   R   I
X.19     2  .   Q   A   N   P   F   B   D   J   L   I   K   C   E   R   O   G
X.20     2  .   R   L   F   M   P   I   E   Q   O   J   B   H   N   K   C   A

      2   .   .
     37   1   1

        37q 37r
     2P 37c 37a
     3P 37n 37q
     5P 37k 37p
     7P 37h 37o
    11P 37b 37m
    13P 37a 37l
    17P 37g 37j
    19P 37j 37i
    23P 37p 37g
    29P 37l 37d
    31P 37i 37c
    37P  1a  1a

X.1       1   1
X.2       1   1
X.3       N   Q
X.4       L   N
X.5       M   B
X.6       J   O
X.7       Q   C
X.8       K   G
X.9       E   J
X.10      D   I
X.11      O   F
X.12      C   P
X.13      A   E
X.14      F   R
X.15      I   L
X.16      R   H
X.17      P   K
X.18      B   A
X.19      H   M
X.20      G   D

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