Show commands:
Magma
magma: G := TransitiveGroup(35, 18);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $D_7\times A_5$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,8,5,7,3,10)(2,6)(4,9)(11,33)(12,35,14,32,13,31)(15,34)(16,26,20,29,18,28)(17,27)(19,30)(22,24,25), (1,15,23,35,6,17,28,2,11,24,34,9,20,30,4,13,21,33,7,19,27)(3,14,25,31,8,16,29)(5,12,22,32,10,18,26) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $14$: $D_{7}$ $60$: $A_5$ $120$: $A_5\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $A_5$
Degree 7: $D_{7}$
Low degree siblings
42T139Siblings are shown 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, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,24, 7,28,13,35,20)( 2,21, 6,30,15,34,19)( 3,25, 8,29,14,31,16) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$ |
| $ 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,29,16, 8,31,25,14) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $105$ | $2$ | $( 1,14)( 2,12)( 3,13)( 4,11)( 5,15)( 6,10)( 7, 8)(16,35)(17,33)(18,34)(19,32) (20,31)(21,26)(22,30)(23,27)(24,29)(25,28)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,29,20, 8,35,25,13, 3,28,16, 7,31,24,14)( 2,26,19,10,34,22,15, 5,30,18, 6, 32,21,12)( 4,27,17, 9,33,23,11)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,25, 7,29,13,31,20, 3,24, 8,28,14,35,16)( 2,22, 6,26,15,32,19, 5,21,10,30, 12,34,18)( 4,23, 9,27,11,33,17)$ |
| $ 14, 14, 7 $ | $30$ | $14$ | $( 1,31,28,25,20,14, 7, 3,35,29,24,16,13, 8)( 2,32,30,22,19,12, 6, 5,34,26,21, 18,15,10)( 4,33,27,23,17,11, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 3)( 2, 5)( 6,10)( 7, 8)(12,15)(13,14)(16,20)(18,19)(21,22)(24,25)(26,30) (28,29)(31,35)(32,34)$ |
| $ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $ | $140$ | $6$ | $( 1,14, 2,13, 3,15)( 4,11)( 5,12)( 6, 7, 8)(16,34,20,31,19,35)(17,33)(18,32) (21,28,25,30,24,29)(22,26)(23,27)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1,29,19, 7,31,21,13, 3,30,20, 8,34,24,14, 2,28,16, 6,35,25,15) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1,25, 6,28,14,34,20, 3,21, 7,29,15,35,16, 2,24, 8,30,13,31,19) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$ |
| $ 21, 7, 7 $ | $40$ | $21$ | $( 1,31,30,24,16,15, 7, 3,34,28,25,19,13, 8, 2,35,29,21,20,14, 6) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 3, 2)( 6, 7, 8)(13,14,15)(16,19,20)(21,24,25)(28,29,30)(31,34,35)$ |
| $ 10, 10, 10, 5 $ | $84$ | $10$ | $( 1,14, 2,12, 4,13, 3,15, 5,11)( 6,10, 9, 7, 8)(16,34,18,33,20,31,19,32,17,35) (21,26,23,28,25,30,22,27,24,29)$ |
| $ 35 $ | $24$ | $35$ | $( 1,29,19,10,33,24,14, 2,26,17, 7,31,21,12, 4,28,16, 6,32,23,13, 3,30,18, 9, 35,25,15, 5,27,20, 8,34,22,11)$ |
| $ 35 $ | $24$ | $35$ | $( 1,25, 6,26,11,35,16, 2,22, 9,28,14,34,18, 4,24, 8,30,12,33,20, 3,21,10,27, 13,31,19, 5,23, 7,29,15,32,17)$ |
| $ 35 $ | $24$ | $35$ | $( 1,31,30,22,17,13, 8, 2,32,27,24,16,15,10, 4,35,29,21,18,11, 7, 3,34,26,23, 20,14, 6, 5,33,28,25,19,12, 9)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 3, 2, 5, 4)( 6,10, 9, 7, 8)(11,13,14,15,12)(16,19,18,17,20) (21,22,23,24,25)(26,27,28,29,30)(31,34,32,33,35)$ |
| $ 10, 10, 10, 5 $ | $84$ | $10$ | $( 1,14, 2,11, 5,13, 3,15, 4,12)( 6, 9,10, 7, 8)(16,34,17,32,20,31,19,33,18,35) (21,27,22,28,25,30,23,26,24,29)$ |
| $ 35 $ | $24$ | $35$ | $( 1,29,19, 9,32,24,14, 2,27,18, 7,31,21,11, 5,28,16, 6,33,22,13, 3,30,17,10, 35,25,15, 4,26,20, 8,34,23,12)$ |
| $ 35 $ | $24$ | $35$ | $( 1,25, 6,27,12,35,16, 2,23,10,28,14,34,17, 5,24, 8,30,11,32,20, 3,21, 9,26, 13,31,19, 4,22, 7,29,15,33,18)$ |
| $ 35 $ | $24$ | $35$ | $( 1,31,30,23,18,13, 8, 2,33,26,24,16,15, 9, 5,35,29,21,17,12, 7, 3,34,27,22, 20,14, 6, 4,32,28,25,19,11,10)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 3, 2, 4, 5)( 6, 9,10, 7, 8)(11,12,13,14,15)(16,19,17,18,20) (21,23,22,24,25)(26,28,29,30,27)(31,34,33,32,35)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $840=2^{3} \cdot 3 \cdot 5 \cdot 7$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | no | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 840.137 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);