Properties

Label 31T2
Order \(62\)
n \(31\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_{31}$

Learn more about

Group action invariants

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

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

Group invariants

Order:  $62=2 \cdot 31$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [62, 1]
Character table:   
      2  1  1   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .
     31  1  .   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

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

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