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Properties

Label 31T4
Degree $31$
Order $155$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{31}:C_{5}$

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Show commands: Magma

magma: G := TransitiveGroup(31, 4);
 

Group action invariants

Degree $n$:  $31$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{31}:C_{5}$
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), (1,16,8,4,2)(3,17,24,12,6)(5,18,9,20,10)(7,19,25,28,14)(11,21,26,13,22)(15,23,27,29,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $155=5 \cdot 31$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  155.1
magma: IdentifyGroup(G);
 
Character table:    
      5  1  1  1  1  1   .   .   .   .   .   .
     31  1  .  .  .  .   1   1   1   1   1   1

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

X.1      1  1  1  1  1   1   1   1   1   1   1
X.2      1  A  B /B /A   1   1   1   1   1   1
X.3      1  B /A  A /B   1   1   1   1   1   1
X.4      1 /B  A /A  B   1   1   1   1   1   1
X.5      1 /A /B  B  A   1   1   1   1   1   1
X.6      5  .  .  .  .   C   E   D  /E  /D  /C
X.7      5  .  .  .  .   D  /C  /E   C   E  /D
X.8      5  .  .  .  .   E   D  /C  /D   C  /E
X.9      5  .  .  .  .  /C  /E  /D   E   D   C
X.10     5  .  .  .  .  /E  /D   C   D  /C   E
X.11     5  .  .  .  .  /D   C   E  /C  /E   D

A = E(5)^4
B = E(5)^3
C = E(31)+E(31)^2+E(31)^4+E(31)^8+E(31)^16
D = E(31)^5+E(31)^9+E(31)^10+E(31)^18+E(31)^20
E = E(31)^3+E(31)^6+E(31)^12+E(31)^17+E(31)^24

magma: CharacterTable(G);