Show commands:
Magma
magma: G := TransitiveGroup(31, 8);
Group action invariants
| Degree $n$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $F_{31}$ | ||
| 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: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31), (1,3,9,27,19,26,16,17,20,29,25,13,8,24,10,30,28,22,4,12,5,15,14,11,2,6,18,23,7,21) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $5$: $C_5$ $6$: $C_6$ $10$: $C_{10}$ $15$: $C_{15}$ $30$: $C_{30}$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
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 $ | $1$ | $1$ | $()$ |
| $ 31 $ | $30$ | $31$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2,17, 9, 5, 3)( 4,18,25,13, 7)( 6,19,10,21,11)( 8,20,26,29,15) (12,22,27,14,23)(16,24,28,30,31)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 9, 3,17, 5)( 4,25, 7,18,13)( 6,10,11,19,21)( 8,26,15,20,29) (12,27,23,22,14)(16,28,31,24,30)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 3, 5, 9,17)( 4, 7,13,25,18)( 6,11,21,10,19)( 8,15,29,26,20) (12,23,14,27,22)(16,31,30,28,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $31$ | $5$ | $( 2, 5,17, 3, 9)( 4,13,18, 7,25)( 6,21,19,11,10)( 8,29,20,15,26) (12,14,22,23,27)(16,30,24,31,28)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,10,20,17,21,26, 9,11,29, 5, 6,15, 3,19, 8)( 4,28,27,18,30,14,25,31,23,13, 16,12, 7,24,22)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,21,29, 3,10,26, 5,19,20, 9, 6, 8,17,11,15)( 4,30,23, 7,28,14,13,24,27,25, 16,22,18,31,12)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,11, 8, 9,19,26, 3,21,15,17, 6,20, 5,10,29)( 4,31,22,25,24,14, 7,30,12,18, 16,27,13,28,23)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,19,15, 5,11,26,17,10, 8, 3, 6,29, 9,21,20)( 4,24,12,13,31,14,18,28,22, 7, 16,23,25,30,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $31$ | $3$ | $( 2, 6,26)( 3,11,20)( 4,16,14)( 5,21, 8)( 7,31,27)( 9,10,15)(12,25,28) (13,30,22)(17,19,29)(18,24,23)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,20,21, 9,29, 6, 3, 8,10,17,26,11, 5,15,19)( 4,27,30,25,23,16, 7,22,28,18, 14,31,13,12,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $31$ | $3$ | $( 2,26, 6)( 3,20,11)( 4,14,16)( 5, 8,21)( 7,27,31)( 9,15,10)(12,28,25) (13,22,30)(17,29,19)(18,23,24)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,29,10, 5,20, 6,17,15,21, 3,26,19, 9, 8,11)( 4,23,28,13,27,16,18,12,30, 7, 14,24,25,22,31)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2, 8,19, 3,15, 6, 5,29,11, 9,26,21,17,20,10)( 4,22,24, 7,12,16,13,23,31,25, 14,30,18,27,28)$ |
| $ 15, 15, 1 $ | $31$ | $15$ | $( 2,15,11,17, 8, 6, 9,20,19, 5,26,10, 3,29,21)( 4,12,31,18,22,16,25,27,24,13, 14,28, 7,23,30)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2, 4,10,28,20,27,17,18,21,30,26,14, 9,25,11,31,29,23, 5,13, 6,16,15,12, 3, 7,19,24, 8,22)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,18,11,16, 8,27, 9,13,19,28,26,23, 3, 4,21,31,15,22,17,25, 6,24,20,14, 5, 7,10,30,29,12)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,25,19,30,15,27, 5, 4,11,24,26,12,17,13,10,31, 8,14, 3,18, 6,28,29,22, 9, 7,21,16,20,23)$ |
| $ 6, 6, 6, 6, 6, 1 $ | $31$ | $6$ | $( 2, 7, 6,31,26,27)( 3,13,11,30,20,22)( 4,19,16,29,14,17)( 5,25,21,28, 8,12) ( 9,18,10,24,15,23)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,13,21,24,29,27, 3,25,10,16,26,22, 5,18,19,31,20,12, 9, 4, 6,30, 8,23,17, 7,11,28,15,14)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,28,17,30, 9,31, 5,16, 3,24)( 4,20,18,26,25,29,13,15, 7, 8)( 6,12,19,22,10, 27,21,14,11,23)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,30, 5,24,17,31, 3,28, 9,16)( 4,26,13, 8,18,29, 7,20,25,15)( 6,22,21,23,19, 27,11,12,10,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $31$ | $2$ | $( 2,31)( 3,30)( 4,29)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,24, 3,16, 5,31, 9,30,17,28)( 4, 8, 7,15,13,29,25,26,18,20)( 6,23,11,14,21, 27,10,22,19,12)$ |
| $ 10, 10, 10, 1 $ | $31$ | $10$ | $( 2,16, 9,28, 3,31,17,24, 5,30)( 4,15,25,20, 7,29,18, 8,13,26)( 6,14,10,12,11, 27,19,23,21,22)$ |
| $ 6, 6, 6, 6, 6, 1 $ | $31$ | $6$ | $( 2,27,26,31, 6, 7)( 3,22,20,30,11,13)( 4,17,14,29,16,19)( 5,12, 8,28,21,25) ( 9,23,15,24,10,18)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,14,15,28,11, 7,17,23, 8,30, 6, 4, 9,12,20,31,19,18, 5,22,26,16,10,25, 3, 27,29,24,21,13)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,23,20,16,21, 7, 9,22,29,28, 6,18, 3,14, 8,31,10,13,17,12,26,24,11, 4, 5, 27,15,30,19,25)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,22, 8,24,19, 7, 3,12,15,16, 6,13, 5,23,29,31,11,25, 9,14,26,30,21,18,17, 27,20,28,10, 4)$ |
| $ 30, 1 $ | $31$ | $30$ | $( 2,12,29,30,10, 7, 5,14,20,24, 6,25,17,22,15,31,21, 4, 3,23,26,28,19,13, 9, 27, 8,16,11,18)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $930=2 \cdot 3 \cdot 5 \cdot 31$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Label: | 930.1 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);