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
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
31T9Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $2$ | $( 2,11)( 3,23)( 4,27)( 5,31)( 6, 8)( 7,12)( 9,24)(10,25)(14,17)(16,29)(20,30) (22,26)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $23250$ | $4$ | $( 2,27,11, 4)( 3, 6,23, 8)( 5,30,31,20)( 7,25,12,10)( 9,17,24,14)(13,18) (16,22,29,26)(19,21)$ | |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $4$ | $( 3,12, 5,16)( 6,10,30,22)( 7,31,29,23)( 8,25,20,26)( 9,14,24,17)(13,19,18,21)$ | |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $4$ | $( 3,16, 5,12)( 6,22,30,10)( 7,23,29,31)( 8,26,20,25)( 9,17,24,14)(13,21,18,19)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $15500$ | $3$ | $( 1,19,18)( 2, 8,16)( 3, 7,14)( 4,31,25)( 5,10,27)( 6,29,11)( 9,20,22) (12,17,23)(13,28,21)(24,30,26)$ | |
| $ 6, 6, 6, 6, 3, 3, 1 $ | $15500$ | $6$ | $( 1,18,19)( 2,29, 8,11,16, 6)( 3,17, 7,23,14,12)( 4,10,31,27,25, 5) ( 9,26,20,24,22,30)(13,21,28)$ | |
| $ 12, 12, 3, 3, 1 $ | $15500$ | $12$ | $( 1,19,18)( 2,31,29,27, 8,25,11, 5,16, 4, 6,10)( 3,22,17,30, 7, 9,23,26,14,20, 12,24)(13,28,21)$ | |
| $ 12, 12, 3, 3, 1 $ | $15500$ | $12$ | $( 1,19,18)( 2, 5,29, 4, 8,10,11,31,16,27, 6,25)( 3,26,17,20, 7,24,23,22,14,30, 12, 9)(13,28,21)$ | |
| $ 8, 8, 8, 2, 2, 2, 1 $ | $15500$ | $8$ | $( 1,13)( 2,12, 4,26,11, 7,27,22)( 3,10,20,16,23,25,30,29)( 5, 9, 8,17,31,24, 6,14)(18,21)(19,28)$ | |
| $ 8, 8, 8, 2, 2, 2, 1 $ | $15500$ | $8$ | $( 1,13)( 2,22,27, 7,11,26, 4,12)( 3,29,30,25,23,16,20,10)( 5,14, 6,24,31,17, 8, 9)(18,21)(19,28)$ | |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,21,19,13,18,28)( 2,30, 5,12,29, 9, 4, 3, 8,26,10,17,11,20,31, 7,16,24,27, 23, 6,22,25,14)$ | |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,21,19,13,18,28)( 2, 3,31,22,29,17,27,30, 8, 7,25, 9,11,23, 5,26,16,14, 4, 20, 6,12,10,24)$ | |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,21,19,13,18,28)( 2,20, 5, 7,29,24, 4,23, 8,22,10,14,11,30,31,12,16, 9,27, 3, 6,26,25,17)$ | |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,21,19,13,18,28)( 2,23,31,26,29,14,27,20, 8,12,25,24,11, 3, 5,22,16,17, 4, 30, 6, 7,10, 9)$ | |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $744$ | $5$ | $( 1,27, 4,15, 2)( 3, 9, 8,22,18)( 5,20,13,24,10)( 6,14,21,25,12) ( 7,23,31,29,28)$ | |
| $ 10, 10, 5, 2, 2, 1, 1 $ | $18600$ | $10$ | $( 1, 8,27,22, 4,18,15, 3, 2, 9)( 5,25,20,12,13, 6,24,14,10,21)( 7,29,23,28,31) (16,30)(17,26)$ | |
| $ 20, 5, 4, 1, 1 $ | $18600$ | $20$ | $( 1,14, 8,10,27,21,22, 5, 4,25,18,20,15,12, 3,13, 2, 6, 9,24)( 7,28,29,31,23) (16,17,30,26)$ | |
| $ 20, 5, 4, 1, 1 $ | $18600$ | $20$ | $( 1,20, 8,12,27,13,22, 6, 4,24,18,14,15,10, 3,21, 2, 5, 9,25)( 7,28,29,31,23) (16,26,30,17)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,14, 4,11,21,24,10, 7,27, 6,16,13,17,19,28, 9,23,29,26,18,15,12,31, 5, 2, 25,22,20, 8, 3,30)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,23,14,29, 4,26,11,18,21,15,24,12,10,31, 7, 5,27, 2, 6,25,16,22,13,20,17, 8,19, 3,28,30, 9)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,27,23, 2,14, 6,29,25, 4,16,26,22,11,13,18,20,21,17,15, 8,24,19,12, 3,10, 28,31,30, 7, 9, 5)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,21,27,17,23,15, 2, 8,14,24, 6,19,29,12,25, 3, 4,10,16,28,26,31,22,30,11, 7,13, 9,18, 5,20)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1, 4,21,10,27,16,17,28,23,26,15,31, 2,22, 8,30,14,11,24, 7, 6,13,19, 9,29, 18,12, 5,25,20, 3)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,30, 3, 8,20,22,25, 2, 5,31,12,15,18,26,29,23, 9,28,19,17,13,16, 6,27, 7, 10,24,21,11, 4,14)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1, 9,30,28, 3,19, 8,17,20,13,22,16,25, 6, 2,27, 5, 7,31,10,12,24,15,21,18, 11,26, 4,29,14,23)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1, 5, 9, 7,30,31,28,10, 3,12,19,24, 8,15,17,21,20,18,13,11,22,26,16, 4,25, 29, 6,14, 2,23,27)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1,20, 5,18, 9,13, 7,11,30,22,31,26,28,16,10, 4, 3,25,12,29,19, 6,24,14, 8, 2,15,23,17,27,21)$ | |
| $ 31 $ | $12000$ | $31$ | $( 1, 3,20,25, 5,12,18,29, 9,19,13, 6, 7,24,11,14,30, 8,22, 2,31,15,26,23,28, 17,16,27,10,21, 4)$ | |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $14880$ | $5$ | $( 1,30, 3, 5, 6)( 2,19, 9,24,21)( 4,17,22,10,14)( 7,29,23,28,31) ( 8,20,12,15,16)(13,25,27,26,18)$ |
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);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 372000.a | magma: IdentifyGroup(G);
| |
| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 31 P | |
| Type |
magma: CharacterTable(G);