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Properties

Label 31T3
Degree $31$
Order $93$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{31}:C_{3}$

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Show commands: Magma

magma: G := TransitiveGroup(31, 3);
 

Group action invariants

Degree $n$:  $31$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{31}:C_{3}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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), (1,25,5)(2,19,10)(3,13,15)(4,7,20)(6,26,30)(8,14,9)(11,27,24)(12,21,29)(16,28,18)(17,22,23)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $93=3 \cdot 31$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  93.1
magma: IdentifyGroup(G);
 
Character table:    
      3  1  1  1   .   .   .   .   .   .   .   .   .   .
     31  1  .  .   1   1   1   1   1   1   1   1   1   1

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(31)^2+E(31)^10+E(31)^19
C = E(31)^17+E(31)^22+E(31)^23
D = E(31)^3+E(31)^13+E(31)^15
E = E(31)^6+E(31)^26+E(31)^30
F = E(31)^4+E(31)^7+E(31)^20

magma: CharacterTable(G);