Properties

Label 31T5
Order \(186\)
n \(31\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{31}:C_{6}$

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Group action invariants

Degree $n$ :  $31$
Transitive number $t$ :  $5$
Group :  $C_{31}:C_{6}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,26,25,30,5,6)(2,21,19,29,10,12)(3,16,13,28,15,18)(4,11,7,27,20,24)(8,22,14,23,9,17), (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:  $C_2$
3:  $C_3$
6:  $C_6$

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

Group invariants

Order:  $186=2 \cdot 3 \cdot 31$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [186, 1]
Character table:   
      2  1  1   1  1   1  1   .   .   .   .   .
      3  1  1   1  1   1  1   .   .   .   .   .
     31  1  .   .  .   .  .   1   1   1   1   1

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

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

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