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Magma
magma: G := TransitiveGroup(31, 10);
Group action invariants
| Degree $n$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PSL(5,2)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| 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);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
31T10Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
| Label | 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 $ | $1$ | $1$ | $()$ | |
| $ 31 $ | $322560$ | $31$ | $( 1, 3,30,26,29,10,13, 4,15,16,19,22,31,18,27,25,12,11,28, 6, 9, 5,24,21, 8, 20,23, 2,14,17, 7)$ | |
| $ 31 $ | $322560$ | $31$ | $( 1,20, 6,18, 4, 3,23, 9,27,15,30, 2, 5,25,16,26,14,24,12,19,29,17,21,11,22, 10, 7, 8,28,31,13)$ | |
| $ 31 $ | $322560$ | $31$ | $( 1,10,19,25, 9,20, 7,29,16,27, 6, 8,17,26,15,18,28,21,14,30, 4,31,11,24, 2, 3,13,22,12, 5,23)$ | |
| $ 31 $ | $322560$ | $31$ | $( 1, 7,17,14, 2,23,20, 8,21,24, 5, 9, 6,28,11,12,25,27,18,31,22,19,16,15, 4, 13,10,29,26,30, 3)$ | |
| $ 31 $ | $322560$ | $31$ | $( 1,13,31,28, 8, 7,10,22,11,21,17,29,19,12,24,14,26,16,25, 5, 2,30,15,27, 9, 23, 3, 4,18, 6,20)$ | |
| $ 31 $ | $322560$ | $31$ | $( 1,23, 5,12,22,13, 3, 2,24,11,31, 4,30,14,21,28,18,15,26,17, 8, 6,27,16,29, 7,20, 9,25,19,10)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $465$ | $2$ | $( 1,30)( 3,21)( 8,14)(11,26)(13,20)(17,18)(19,28)(25,29)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ | $19840$ | $3$ | $( 1,11,17)( 2,31,22)( 3,13, 8)( 5, 7,10)( 6,16,24)( 9,27,12)(14,21,20) (18,30,26)$ | |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $26040$ | $4$ | $( 1, 8,30,14)( 3,26,21,11)( 6, 9)(12,24)(13,18,20,17)(15,23)(16,27) (19,29,28,25)$ | |
| $ 6, 6, 3, 3, 3, 3, 2, 2, 1, 1, 1 $ | $416640$ | $6$ | $( 1,18,11,30,17,26)( 2,22,31)( 3,14,13,21, 8,20)( 5,10, 7)( 6,24,16)( 9,12,27) (19,28)(25,29)$ | |
| $ 12, 6, 4, 3, 3, 2, 1 $ | $833280$ | $12$ | $( 1,21,18, 8,11,20,30, 3,17,14,26,13)( 2,31,22)( 5, 7,10)( 6,27,24, 9,16,12) (15,23)(19,25,28,29)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $6510$ | $2$ | $( 1,11)( 2, 7)( 3, 8)(10,31)(12,24)(13,29)(14,21)(16,27)(17,19)(18,28)(20,25) (26,30)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $312480$ | $4$ | $( 1,13,11,29)( 2,31, 7,10)( 3,28, 8,18)( 5,22)(12,27,24,16)(14,17,21,19) (15,23)(20,26,25,30)$ | |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $78120$ | $4$ | $( 1,14,30, 8)( 2, 7)( 3,17)( 5,10,22,31)( 6,12, 9,24)(11,20)(13,26)(15,23) (18,21)(19,29,28,25)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $55552$ | $3$ | $( 1,12,20)( 2,19, 3)( 4,22, 5)( 7,17, 8)( 9,15,23)(10,18,21)(11,24,25) (13,26,27)(14,31,28)(16,29,30)$ | |
| $ 6, 6, 6, 6, 3, 3, 1 $ | $833280$ | $6$ | $( 1,13,12,26,20,27)( 2,14,19,31, 3,28)( 4, 5,22)( 7,21,17,10, 8,18)( 9,23,15) (11,29,24,30,25,16)$ | |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $666624$ | $5$ | $( 1,27,22,16,11)( 2,28,23,18, 7)( 3,31,15,10, 8)( 4,30,25,20,26) ( 5,29,24,12,13)( 9,21,17,19,14)$ | |
| $ 15, 15, 1 $ | $666624$ | $15$ | $( 1, 5,25,27,29,20,22,24,26,16,12, 4,11,13,30)( 2, 9, 8,28,21, 3,23,17,31,18, 19,15, 7,14,10)$ | |
| $ 15, 15, 1 $ | $666624$ | $15$ | $( 1, 4,24,27,30,12,22,25,13,16,20, 5,11,26,29)( 2,15,17,28,10,19,23, 8,14,18, 3, 9, 7,31,21)$ | |
| $ 7, 7, 7, 7, 1, 1, 1 $ | $238080$ | $7$ | $( 1, 3,19,14,25,20,17)( 2,22,15,10, 6,12,16)( 5,23,31, 9,24,27, 7) ( 8,29,13,18,30,21,28)$ | |
| $ 7, 7, 7, 7, 1, 1, 1 $ | $238080$ | $7$ | $( 1,17,20,25,14,19, 3)( 2,16,12, 6,10,15,22)( 5, 7,27,24, 9,31,23) ( 8,28,21,30,18,13,29)$ | |
| $ 14, 7, 7, 2, 1 $ | $714240$ | $14$ | $( 1,29, 3,13,19,18,14,30,25,21,20,28,17, 8)( 2, 6,22,12,15,16,10) ( 5,24,23,27,31, 7, 9)(11,26)$ | |
| $ 14, 7, 7, 2, 1 $ | $714240$ | $14$ | $( 1,13,14,21,17,29,19,30,20, 8, 3,18,25,28)( 2,12,10,22,16, 6,15) ( 5,27, 9,23, 7,24,31)(11,26)$ | |
| $ 21, 7, 3 $ | $476160$ | $21$ | $( 1,22,30,25,14,15, 6,21,31,13,18,29,10, 4,20,23,19,26,11,27, 2) ( 3, 8,17, 5,24,28,16)( 7,12, 9)$ | |
| $ 21, 7, 3 $ | $476160$ | $21$ | $( 1,18, 2,13,27,31,11,21,26, 6,19,15,23,14,20,25, 4,30,10,22,29) ( 3, 5,16,17,28, 8,24)( 7,12, 9)$ | |
| $ 8, 8, 4, 4, 4, 2, 1 $ | $624960$ | $8$ | $( 1,20, 8,28,30,13,14,19)( 2,12,31,16)( 3,18,26,29,21,17,11,25)( 5,22) ( 6,23, 9,15)( 7,24,10,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $9999360=2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 9999360.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| 31 P | |
| Type |
magma: CharacterTable(G);