Properties

Label 31T7
Order \(465\)
n \(31\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{31}:C_{15}$

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
5:  $C_5$
15:  $C_{15}$

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

Group invariants

Order:  $465=3 \cdot 5 \cdot 31$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [465, 1]
Character table:   
      3  1  1  1  1   1  1   1   1   1  1   1   1   1  1   1   .   .
      5  1  1  1  1   1  1   1   1   1  1   1   1   1  1   1   .   .
     31  1  .  .  .   .  .   .   .   .  .   .   .   .  .   .   1   1

        1a 5a 5b 3a 15a 5c 15b 15c 15d 5d 15e 15f 15g 3b 15h 31a 31b
     2P 1a 5b 5d 3b 15e 5a 15f 15a 15c 5c 15d 15g 15h 3a 15b 31a 31b
     3P 1a 5c 5a 1a  5a 5d  5d  5c  5d 5b  5b  5c  5a 1a  5b 31b 31a
     5P 1a 1a 1a 3b  3a 1a  3b  3b  3a 1a  3b  3a  3b 3a  3a 31a 31b
     7P 1a 5b 5d 3a 15h 5a 15c 15g 15f 5c 15b 15a 15e 3b 15d 31a 31b
    11P 1a 5a 5b 3b 15g 5c 15d 15f 15b 5d 15h 15c 15a 3a 15e 31b 31a
    13P 1a 5c 5a 3a 15f 5d 15e 15b 15h 5b 15g 15d 15c 3b 15a 31b 31a
    17P 1a 5b 5d 3b 15e 5a 15f 15a 15c 5c 15d 15g 15h 3a 15b 31b 31a
    19P 1a 5d 5c 3a 15d 5b 15g 15e 15a 5a 15c 15h 15b 3b 15f 31a 31b
    23P 1a 5c 5a 3b 15c 5d 15h 15d 15e 5b 15a 15b 15f 3a 15g 31b 31a
    29P 1a 5d 5c 3b 15b 5b 15a 15h 15g 5a 15f 15e 15d 3a 15c 31b 31a
    31P 1a 5a 5b 3a 15a 5c 15b 15c 15d 5d 15e 15f 15g 3b 15h  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  C  /C  1   C   C  /C  1   C  /C   C /C  /C   1   1
X.3      1  1  1 /C   C  1  /C  /C   C  1  /C   C  /C  C   C   1   1
X.4      1  A  B  1   B /B  /B   A  /B /A  /A   A   B  1  /A   1   1
X.5      1  B /A  1  /A  A   A   B   A /B  /B   B  /A  1  /B   1   1
X.6      1 /B  A  1   A /A  /A  /B  /A  B   B  /B   A  1   B   1   1
X.7      1 /A /B  1  /B  B   B  /A   B  A   A  /A  /B  1   A   1   1
X.8      1  A  B  C   D /B  /D  /F  /E /A   G  /G   E /C   F   1   1
X.9      1  A  B /C   E /B  /E  /G  /D /A   F  /F   D  C   G   1   1
X.10     1  B /A  C   F  A  /F   E  /G /B  /D   D   G /C  /E   1   1
X.11     1  B /A /C   G  A  /G   D  /F /B  /E   E   F  C  /D   1   1
X.12     1 /B  A  C  /G /A   G  /D   F  B   E  /E  /F /C   D   1   1
X.13     1 /B  A /C  /F /A   F  /E   G  B   D  /D  /G  C   E   1   1
X.14     1 /A /B  C  /E  B   E   G   D  A  /F   F  /D /C  /G   1   1
X.15     1 /A /B /C  /D  B   D   F   E  A  /G   G  /E  C  /F   1   1
X.16    15  .  .  .   .  .   .   .   .  .   .   .   .  .   .   H  /H
X.17    15  .  .  .   .  .   .   .   .  .   .   .   .  .   .  /H   H

A = E(5)^4
B = E(5)^3
C = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
D = E(15)^14
E = E(15)^4
F = E(15)^8
G = E(15)^13
H = E(31)+E(31)^2+E(31)^4+E(31)^5+E(31)^7+E(31)^8+E(31)^9+E(31)^10+E(31)^14+E(31)^16+E(31)^18+E(31)^19+E(31)^20+E(31)^25+E(31)^28
  = (-1+Sqrt(-31))/2 = b31