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Properties

Label 31T7
Degree $31$
Order $465$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{31}:C_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(31, 7);
 

Group action invariants

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

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $465=3 \cdot 5 \cdot 31$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  465.1
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 5A1 5A-1 5A2 5A-2 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 31A1 31A-1
Size 1 31 31 31 31 31 31 31 31 31 31 31 31 31 31 15 15
3 P 1A 3A-1 3A1 5A-2 5A-1 5A2 5A1 15A2 15A-2 15A4 15A-4 15A-1 15A-7 15A7 15A1 31A1 31A-1
5 P 1A 1A 1A 5A2 5A1 5A-2 5A-1 5A1 5A-1 5A2 5A-2 5A2 5A-1 5A1 5A-2 31A-1 31A1
31 P 1A 3A-1 3A1 1A 1A 1A 1A 3A1 3A-1 3A-1 3A1 3A1 3A1 3A-1 3A-1 31A1 31A-1
Type
465.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
465.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
465.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
465.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 1 1
465.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 1 1
465.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 1 1
465.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 1 1
465.1.1d1 C 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154 1 1
465.1.1d2 C 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4 1 1
465.1.1d3 C 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15 1 1
465.1.1d4 C 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1 1 1
465.1.1d5 C 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157 1 1
465.1.1d6 C 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7 1 1
465.1.1d7 C 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2 1 1
465.1.1d8 C 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152 1 1
465.1.15a1 C 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ31−15−ζ31−13−ζ31−12−ζ31−11−ζ31−6−ζ31−3−1−ζ31−ζ312−ζ314−ζ315−ζ317−ζ318−ζ319−ζ3110−ζ3114 ζ31−15+ζ31−13+ζ31−12+ζ31−11+ζ31−6+ζ31−3+ζ31+ζ312+ζ314+ζ315+ζ317+ζ318+ζ319+ζ3110+ζ3114
465.1.15a2 C 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ31−15+ζ31−13+ζ31−12+ζ31−11+ζ31−6+ζ31−3+ζ31+ζ312+ζ314+ζ315+ζ317+ζ318+ζ319+ζ3110+ζ3114 −ζ31−15−ζ31−13−ζ31−12−ζ31−11−ζ31−6−ζ31−3−1−ζ31−ζ312−ζ314−ζ315−ζ317−ζ318−ζ319−ζ3110−ζ3114

magma: CharacterTable(G);