# Properties

 Label 31T10 Degree $31$ Order $9999360$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(5,2)$

Show commands: Magma

magma: G := TransitiveGroup(31, 10);

## Group action invariants

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

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

31T10

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $465$ $2$ $( 1,10)( 3,16)( 4,18)( 7,11)( 8,24)( 9,29)(15,20)(19,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1$ $19840$ $3$ $( 2, 5,30)( 3,29,24)( 4,19,11)( 6,27,13)( 7,18,22)( 8,16, 9)(21,23,25) (26,28,31)$ $4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $26040$ $4$ $( 1,15,10,20)( 3,19,16,22)( 4, 8,18,24)( 6,25)( 7,29,11, 9)(12,14)(13,23) (21,27)$ $6, 6, 3, 3, 3, 3, 2, 2, 1, 1, 1$ $416640$ $6$ $( 1,10)( 2,30, 5)( 3, 8,29,16,24, 9)( 4, 7,19,18,11,22)( 6,13,27)(15,20) (21,25,23)(26,31,28)$ $12, 6, 4, 3, 3, 2, 1$ $833280$ $12$ $( 1,20,10,15)( 2, 5,30)( 3, 7, 8,19,29,18,16,11,24,22, 9, 4)( 6,21,13,25,27,23 )(12,14)(26,28,31)$ $7, 7, 7, 7, 1, 1, 1$ $238080$ $7$ $( 1,26,23, 6,11, 9,30)( 2,29, 7,13,20,25,31)( 3, 4,22,21, 5, 8,14) (16,28,27,18,17,19,24)$ $7, 7, 7, 7, 1, 1, 1$ $238080$ $7$ $( 1,30, 9,11, 6,23,26)( 2,31,25,20,13, 7,29)( 3,14, 8, 5,21,22, 4) (16,24,19,17,18,27,28)$ $21, 7, 3$ $476160$ $21$ $( 1,27, 2,26,18,29,23,17, 7, 6,19,13,11,24,20, 9,16,25,30,28,31) ( 3, 8,21, 4,14, 5,22)(10,15,12)$ $21, 7, 3$ $476160$ $21$ $( 1,19,31, 6,28, 7,30,17,25,23,16,29, 9,18,20,26,24, 2,11,27,13) ( 3, 4,22,21, 5, 8,14)(10,15,12)$ $14, 7, 7, 2, 1$ $714240$ $14$ $( 1, 6,30,23, 9,26,11)( 2,18,31,27,25,28,20,16,13,24, 7,19,29,17) ( 3,21,14,22, 8, 4, 5)(10,12)$ $14, 7, 7, 2, 1$ $714240$ $14$ $( 1,23,11,30,26, 6, 9)( 2,27,20,24,29,18,25,16, 7,17,31,28,13,19) ( 3,22, 5,14, 4,21, 8)(10,12)$ $31$ $322560$ $31$ $( 1,14,29, 5,22,27,21,13,12, 8,26,28,10,25,16,19,23, 6,15, 7,20, 4,24,17, 2, 30,18, 9,31,11, 3)$ $31$ $322560$ $31$ $( 1,30, 7,25,13,14,18,20,16,12,29, 9, 4,19, 8, 5,31,24,23,26,22,11,17, 6,28, 27, 3, 2,15,10,21)$ $31$ $322560$ $31$ $( 1,27,26,19,20,30, 3,22, 8,16, 7, 2,11, 5,12,25,15,17,31,29,13,10, 6,24, 9, 14,21,28,23, 4,18)$ $31$ $322560$ $31$ $( 1, 3,11,31, 9,18,30, 2,17,24, 4,20, 7,15, 6,23,19,16,25,10,28,26, 8,12,13, 21,27,22, 5,29,14)$ $31$ $322560$ $31$ $( 1,21,10,15, 2, 3,27,28, 6,17,11,22,26,23,24,31, 5, 8,19, 4, 9,29,12,16,20, 18,14,13,25, 7,30)$ $31$ $322560$ $31$ $( 1,18, 4,23,28,21,14, 9,24, 6,10,13,29,31,17,15,25,12, 5,11, 2, 7,16, 8,22, 3,30,20,19,26,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $6510$ $2$ $( 1, 8)( 2,27)( 3,30)( 4, 5)( 6,15)( 7,12)(10,24)(11,14)(16,31)(18,28)(20,25) (21,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $55552$ $3$ $( 1, 6,12)( 2,20, 3)( 4,31,26)( 5,16,21)( 7, 8,15)( 9,23,22)(10,28,11) (13,29,17)(14,24,18)(25,30,27)$ $6, 6, 6, 6, 3, 3, 1$ $833280$ $6$ $( 1, 7, 6, 8,12,15)( 2,30,20,27, 3,25)( 4,21,31, 5,26,16)( 9,22,23) (10,14,28,24,11,18)(13,17,29)$ $4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1$ $312480$ $4$ $( 1,26, 8,21)( 2,24,27,10)( 3, 6,14, 4)( 5,30,15,11)( 7,28,31,20)( 9,22) (12,18,16,25)(19,29)$ $5, 5, 5, 5, 5, 5, 1$ $666624$ $5$ $( 1, 8, 4,13, 5)( 2,27,28, 9,18)( 3,30,10,22,24)( 6,15,31,29,16) ( 7,26,17,21,12)(11,23,14,20,25)$ $15, 15, 1$ $666624$ $15$ $( 1,26,16, 8,17, 6, 4,21,15,13,12,31, 5, 7,29)( 2,10,14,27,22,20,28,24,25, 9, 3,11,18,30,23)$ $15, 15, 1$ $666624$ $15$ $( 1,31,21, 8,29,12, 4,16, 7,13, 6,26, 5,15,17)( 2,11,24,27,23, 3,28,14,30, 9, 20,10,18,25,22)$ $4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $78120$ $4$ $( 1,21,30, 3)( 2,22, 7, 5)( 6, 9)( 8,19)(10,31)(11,13)(12,23,24,15)(14,28) (17,25,18,29)(20,26)$ $8, 8, 4, 4, 4, 2, 1$ $624960$ $8$ $( 1,15,21,12,30,23, 3,24)( 2,17,22,25, 7,18, 5,29)( 6,31, 9,10)( 8,26,19,20) (11,28,13,14)(16,27)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $9999360=2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 9999360.a magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);