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Properties

Label 31T9
Degree $31$
Order $372000$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(3,5)$

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Show commands: Magma

magma: G := TransitiveGroup(31, 9);
 

Group action invariants

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

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

31T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 31 $ $12000$ $31$ $( 1,14,17,22, 2, 8, 3,28,25,19,21,11,16, 4,27,13, 6,30,31,15,23,29,10, 9, 5, 20, 7,18,12,26,24)$
$ 31 $ $12000$ $31$ $( 1, 6,14,30,17,31,22,15, 2,23, 8,29, 3,10,28, 9,25, 5,19,20,21, 7,11,18,16, 12, 4,26,27,24,13)$
$ 31 $ $12000$ $31$ $( 1,25, 6, 5,14,19,30,20,17,21,31, 7,22,11,15,18, 2,16,23,12, 8, 4,29,26, 3, 27,10,24,28,13, 9)$
$ 31 $ $12000$ $31$ $( 1, 2,25,16, 6,23, 5,12,14, 8,19, 4,30,29,20,26,17, 3,21,27,31,10, 7,24,22, 28,11,13,15, 9,18)$
$ 31 $ $12000$ $31$ $( 1,17, 2, 3,25,21,16,27, 6,31,23,10, 5, 7,12,24,14,22, 8,28,19,11, 4,13,30, 15,29, 9,20,18,26)$
$ 31 $ $12000$ $31$ $( 1,24,26,12,18, 7,20, 5, 9,10,29,23,15,31,30, 6,13,27, 4,16,11,21,19,25,28, 3, 8, 2,22,17,14)$
$ 31 $ $12000$ $31$ $( 1,13,24,27,26, 4,12,16,18,11, 7,21,20,19, 5,25, 9,28,10, 3,29, 8,23, 2,15, 22,31,17,30,14, 6)$
$ 31 $ $12000$ $31$ $( 1, 9,13,28,24,10,27, 3,26,29, 4, 8,12,23,16, 2,18,15,11,22, 7,31,21,17,20, 30,19,14, 5, 6,25)$
$ 31 $ $12000$ $31$ $( 1,18, 9,15,13,11,28,22,24, 7,10,31,27,21, 3,17,26,20,29,30, 4,19, 8,14,12, 5,23, 6,16,25, 2)$
$ 31 $ $12000$ $31$ $( 1,26,18,20, 9,29,15,30,13, 4,11,19,28, 8,22,14,24,12, 7, 5,10,23,31, 6,27, 16,21,25, 3, 2,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $775$ $2$ $( 1,31)( 4, 9)( 6,22)( 7,21)( 8,17)(11,13)(14,26)(15,25)(18,29)(19,23)(20,24) (27,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $15500$ $3$ $( 1,19,20)( 2,16,28)( 3,12, 5)( 4,11,26)( 6,21,29)( 7,18,22)( 8,15,30) ( 9,13,14)(17,25,27)(23,24,31)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $775$ $4$ $( 1,26,31,14)( 4,23, 9,19)( 6,30,22,27)( 7,17,21, 8)(11,24,13,20)(15,18,25,29)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $775$ $4$ $( 1,14,31,26)( 4,19, 9,23)( 6,27,22,30)( 7, 8,21,17)(11,20,13,24)(15,29,25,18)$
$ 6, 6, 6, 6, 3, 3, 1 $ $15500$ $6$ $( 1,24,19,31,20,23)( 2,28,16)( 3, 5,12)( 4,14,11, 9,26,13)( 6,18,21,22,29, 7) ( 8,27,15,17,30,25)$
$ 8, 8, 8, 2, 2, 2, 1 $ $15500$ $8$ $( 1, 7,26,17,31,21,14, 8)( 2, 3)( 4,25,23,29, 9,15,19,18)( 5,28) ( 6,13,30,20,22,11,27,24)(12,16)$
$ 8, 8, 8, 2, 2, 2, 1 $ $15500$ $8$ $( 1, 8,14,21,31,17,26, 7)( 2, 3)( 4,18,19,15, 9,29,23,25)( 5,28) ( 6,24,27,11,22,20,30,13)(12,16)$
$ 12, 12, 3, 3, 1 $ $15500$ $12$ $( 1, 9,24,26,19,13,31, 4,20,14,23,11)( 2,16,28)( 3,12, 5)( 6,17,18,30,21,25, 22, 8,29,27, 7,15)$
$ 12, 12, 3, 3, 1 $ $15500$ $12$ $( 1, 4,24,14,19,11,31, 9,20,26,23,13)( 2,16,28)( 3,12, 5)( 6, 8,18,27,21,15, 22,17,29,30, 7,25)$
$ 24, 6, 1 $ $15500$ $24$ $( 1,27, 9, 7,24,15,26, 6,19,17,13,18,31,30, 4,21,20,25,14,22,23, 8,11,29) ( 2, 5,16, 3,28,12)$
$ 24, 6, 1 $ $15500$ $24$ $( 1, 6, 4, 8,24,18,14,27,19,21,11,15,31,22, 9,17,20,29,26,30,23, 7,13,25) ( 2, 5,16, 3,28,12)$
$ 24, 6, 1 $ $15500$ $24$ $( 1,30, 9,21,24,25,26,22,19, 8,13,29,31,27, 4, 7,20,15,14, 6,23,17,11,18) ( 2, 5,16, 3,28,12)$
$ 24, 6, 1 $ $15500$ $24$ $( 1,22, 4,17,24,29,14,30,19, 7,11,25,31, 6, 9, 8,20,18,26,27,23,21,13,15) ( 2, 5,16, 3,28,12)$
$ 5, 5, 5, 5, 5, 5, 1 $ $14880$ $5$ $( 1,17,24,20, 3)( 4, 8,18,19,16)( 5,15,10,23,14)( 6,29,12,27, 7) ( 9,13,30,31,25)(11,21,26,22,28)$
$ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1 $ $744$ $5$ $( 1,19,18,13,21)( 2, 3,12,16, 5)( 4,25,20, 8,26)( 7,31,23,29,11) ( 9,15,24,17,14)$
$ 10, 10, 5, 2, 2, 1, 1 $ $18600$ $10$ $( 1,26,19, 4,18,25,13,20,21, 8)( 2,24, 3,17,12,14,16, 9, 5,15)( 6,27) ( 7,29,31,11,23)(10,30)$
$ 20, 5, 4, 1, 1 $ $18600$ $20$ $( 1,24,26, 3,19,17, 4,12,18,14,25,16,13, 9,20, 5,21,15, 8, 2)( 6,10,27,30) ( 7,11,29,23,31)$
$ 20, 5, 4, 1, 1 $ $18600$ $20$ $( 1,16,26, 9,19, 5, 4,15,18, 2,25,24,13, 3,20,17,21,12, 8,14)( 6,30,27,10) ( 7,11,29,23,31)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ $23250$ $4$ $( 1,15,31,25)( 2, 5)( 3,16)( 4,19, 9,23)( 6,30,22,27)( 7,20,21,24) ( 8,13,17,11)(14,29,26,18)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $372000=2^{5} \cdot 3 \cdot 5^{3} \cdot 31$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  372000.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);