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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);
 
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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{31}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{7}$ $775$ $2$ $12$ $( 1, 3)( 2, 8)( 4, 9)( 5,24)( 7,17)(13,20)(15,18)(16,29)(19,23)(22,27)(26,31)(28,30)$
3A $3^{10},1$ $15500$ $3$ $20$ $( 1, 4,25)( 2,28,17)( 3, 9,21)( 5,15, 8)( 6,31,19)( 7,16,24)(10,12,14)(11,26,23)(18,29,30)(20,22,27)$
4A1 $4^{6},1^{7}$ $775$ $4$ $18$ $( 1,26, 3,31)( 2,17, 8, 7)( 4,19, 9,23)( 5,20,24,13)(15,16,18,29)(22,28,27,30)$
4A-1 $4^{6},1^{7}$ $775$ $4$ $18$ $( 1,31, 3,26)( 2, 7, 8,17)( 4,23, 9,19)( 5,13,24,20)(15,29,18,16)(22,30,27,28)$
4B $4^{6},2^{2},1^{3}$ $23250$ $4$ $20$ $( 1,16, 3,29)( 2,17, 8, 7)( 4,30, 9,28)( 5,13,24,20)( 6,12)(14,25)(15,26,18,31)(19,22,23,27)$
5A $5^{5},1^{6}$ $744$ $5$ $20$ $( 1, 7,20,16, 9)( 2,23,24,26,18)( 3, 4,29,13,17)( 5,19, 8,15,31)(10,30,22,27,28)$
5B $5^{6},1$ $14880$ $5$ $24$ $( 1,28, 4, 5,24)( 2,16,22,17,15)( 3,31,23, 9,27)( 7,10,29,19,26)( 8,18,20,30,13)(11,12,21,25,14)$
6A $6^{4},3^{2},1$ $15500$ $6$ $24$ $( 1, 9, 4,21,25, 3)( 2,29,28,30,17,18)( 5, 8,15)( 6,19,31)( 7,10,16,12,24,14)(11,27,26,20,23,22)$
8A1 $8^{3},2^{3},1$ $15500$ $8$ $24$ $( 1,23,26, 4, 3,19,31, 9)( 2,20,17,24, 8,13, 7, 5)( 6,25)(11,21)(12,14)(15,30,16,22,18,28,29,27)$
8A-1 $8^{3},2^{3},1$ $15500$ $8$ $24$ $( 1, 4,31,23, 3, 9,26,19)( 2,24, 7,20, 8, 5,17,13)( 6,25)(11,21)(12,14)(15,22,29,30,18,27,16,28)$
10A $10^{2},5,2^{2},1^{2}$ $18600$ $10$ $24$ $( 1,26, 7,18,20, 2,16,23, 9,24)( 3,13, 4,17,29)( 5,27,19,28, 8,10,15,30,31,22)(12,14)(21,25)$
12A1 $12^{2},3^{2},1$ $15500$ $12$ $26$ $( 1,29, 9,28, 4,30,21,17,25,18, 3, 2)( 5,15, 8)( 6,31,19)( 7,23,10,22,16,11,12,27,24,26,14,20)$
12A-1 $12^{2},3^{2},1$ $15500$ $12$ $26$ $( 1,17, 9,18, 4, 2,21,29,25,28, 3,30)( 5,15, 8)( 6,31,19)( 7,27,10,26,16,20,12,23,24,22,14,11)$
20A1 $20,5,4,1^{2}$ $18600$ $20$ $26$ $( 1,22,26, 5, 7,27,18,19,20,28, 2, 8,16,10,23,15, 9,30,24,31)( 3,17,13,29, 4)(12,21,14,25)$
20A-1 $20,5,4,1^{2}$ $18600$ $20$ $26$ $( 1, 5,18,28,16,15,24,22, 7,19, 2,10, 9,31,26,27,20, 8,23,30)( 3,29,17, 4,13)(12,25,14,21)$
24A1 $24,6,1$ $15500$ $24$ $28$ $( 1,22,29,16, 9,11,28,12, 4,27,30,24,21,26,17,14,25,20,18, 7, 3,23, 2,10)( 5,19,15, 6, 8,31)$
24A-1 $24,6,1$ $15500$ $24$ $28$ $( 1,26,29,14, 9,20,28, 7, 4,23,30,10,21,22,17,16,25,11,18,12, 3,27, 2,24)( 5,19,15, 6, 8,31)$
24A7 $24,6,1$ $15500$ $24$ $28$ $( 1, 7,17,27, 9,10,18,26, 4,16, 2,20,21,12,29,23,25,24,28,22, 3,14,30,11)( 5,19,15, 6, 8,31)$
24A-7 $24,6,1$ $15500$ $24$ $28$ $( 1,12,17,23, 9,24,18,22, 4,14, 2,11,21, 7,29,27,25,10,28,26, 3,16,30,20)( 5,19,15, 6, 8,31)$
31A1 $31$ $12000$ $31$ $30$ $( 1,24,19,17,12, 7,11,18,15,23, 8,25,30,27,26,31,22,29, 3,16,20,13,28, 9,10,21,14, 4, 2, 6, 5)$
31A-1 $31$ $12000$ $31$ $30$ $( 1,12,15,30,22,20,10, 2,24, 7,23,27,29,13,21, 6,19,11, 8,26, 3,28,14, 5,17,18,25,31,16, 9, 4)$
31A2 $31$ $12000$ $31$ $30$ $( 1,11,30, 3,10, 5, 7,25,29, 9, 6,12, 8,22,28, 2,17,23,31,13, 4,19,15,26,20,14,24,18,27,16,21)$
31A-2 $31$ $12000$ $31$ $30$ $( 1,30,10, 7,29, 6, 8,28,17,31, 4,15,20,24,27,21,11, 3, 5,25, 9,12,22, 2,23,13,19,26,14,18,16)$
31A3 $31$ $12000$ $31$ $30$ $( 1,10,29, 8,17, 4,20,27,11, 5, 9,22,23,19,14,16,30, 7, 6,28,31,15,24,21, 3,25,12, 2,13,26,18)$
31A-3 $31$ $12000$ $31$ $30$ $( 1,27,14,15,13,17,22, 6,25,10,11,16,24,26, 4,23,28,12,29, 5,30,21,18,20,19,31, 2, 8, 9, 7, 3)$
31A4 $31$ $12000$ $31$ $30$ $( 1,22,24,29,19, 3,17,16,12,20, 7,13,11,28,18, 9,15,10,23,21, 8,14,25, 4,30, 2,27, 6,26, 5,31)$
31A-4 $31$ $12000$ $31$ $30$ $( 1,15,22,10,24,23,29,21,19, 8, 3,14,17,25,16, 4,12,30,20, 2, 7,27,13, 6,11,26,28, 5,18,31, 9)$
31A8 $31$ $12000$ $31$ $30$ $( 1,19,12,11,15, 8,30,26,22, 3,20,28,10,14, 2, 5,24,17, 7,18,23,25,27,31,29,16,13, 9,21, 4, 6)$
31A-8 $31$ $12000$ $31$ $30$ $( 1,29,17,20,11, 9,23,14,30, 6,31,24, 3,12,13,18,10, 8, 4,27, 5,22,19,16, 7,28,15,21,25, 2,26)$

Malle's constant $a(G)$:     $1/12$

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);
 
Nilpotency class:   not nilpotent
Label:  372000.a
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 4A1 4A-1 4B 5A 5B 6A 8A1 8A-1 10A 12A1 12A-1 20A1 20A-1 24A1 24A-1 24A7 24A-7 31A1 31A-1 31A2 31A-2 31A3 31A-3 31A4 31A-4 31A8 31A-8
Size 1 775 15500 775 775 23250 744 14880 15500 15500 15500 18600 15500 15500 18600 18600 15500 15500 15500 15500 12000 12000 12000 12000 12000 12000 12000 12000 12000 12000
2 P 1A 1A 3A 2A 2A 2A 5A 5B 3A 4A1 4A-1 5A 6A 6A 10A 10A 12A1 12A1 12A-1 12A-1 31A8 31A1 31A-8 31A3 31A-1 31A-4 31A4 31A2 31A-3 31A-2
3 P 1A 2A 1A 4A-1 4A1 4B 5A 5B 2A 8A-1 8A1 10A 4A1 4A-1 20A-1 20A1 8A1 8A1 8A-1 8A-1 31A-2 31A-8 31A2 31A4 31A8 31A1 31A-1 31A3 31A-4 31A-3
5 P 1A 2A 3A 4A1 4A-1 4B 1A 1A 6A 8A1 8A-1 2A 12A1 12A-1 4A1 4A-1 24A1 24A-7 24A-1 24A7 31A4 31A-3 31A-4 31A-8 31A3 31A-2 31A2 31A1 31A8 31A-1
31 P 1A 2A 3A 4A-1 4A1 4B 5A 5B 6A 8A-1 8A1 10A 12A-1 12A1 20A-1 20A1 24A7 24A-1 24A-7 24A1 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A
Type

magma: CharacterTable(G);