Show commands:
Magma
magma: G := TransitiveGroup(35, 50);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $A_7:F_5$ | ||
| 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: | (1,35,22,8,16,5,32,23,6,20,2,33,21,10,17,3,31,25,7,18)(4,34,24,9,19)(11,30,12,28)(13,26,15,27)(14,29), (1,8,20,2,9,16,3,10,17,4,6,18,5,7,19)(11,23,15,22,14,21,13,25,12,24)(26,33,30,32,29,31,28,35,27,34) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ $5040$: $S_7$ $10080$: 28T360 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $S_7$
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
| 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, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ | |
| $ 10, 10, 5, 5, 5 $ | $420$ | $10$ | $( 1, 5, 4, 3, 2)( 6,30, 9,28, 7,26,10,29, 8,27)(11,15,14,13,12) (16,35,19,33,17,31,20,34,18,32)(21,25,24,23,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(16,31)(17,32)(18,33)(19,34)(20,35)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $525$ | $2$ | $( 1, 2)( 3, 5)( 6,27)( 7,26)( 8,30)( 9,29)(10,28)(11,12)(13,15)(16,32)(17,31) (18,35)(19,34)(20,33)(21,22)(23,25)$ | |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $3150$ | $4$ | $( 1, 2)( 3, 5)( 6,32,26,17)( 7,31,27,16)( 8,35,28,20)( 9,34,29,19) (10,33,30,18)(11,22)(12,21)(13,25)(14,24)(15,23)$ | |
| $ 20, 10, 5 $ | $2520$ | $20$ | $( 1, 5, 4, 3, 2)( 6,35,29,18, 7,31,30,19, 8,32,26,20, 9,33,27,16,10,34,28,17) (11,25,14,23,12,21,15,24,13,22)$ | |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 6,31,26,16)( 7,32,27,17)( 8,33,28,18)( 9,34,29,19)(10,35,30,20)(11,21) (12,22)(13,23)(14,24)(15,25)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 2, 3, 5, 4)( 6,26)( 7,28,10,29)( 8,30, 9,27)(11,21)(12,23,15,24) (13,25,14,22)(16,31)(17,33,20,34)(18,35,19,32)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $525$ | $4$ | $( 2, 4, 5, 3)( 6,26)( 7,29,10,28)( 8,27, 9,30)(11,21)(12,24,15,23) (13,22,14,25)(16,31)(17,34,20,33)(18,32,19,35)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 2, 4, 5, 3)( 6,16,26,31)( 7,19,30,33)( 8,17,29,35)( 9,20,28,32)(10,18,27,34) (12,14,15,13)(22,24,25,23)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $1050$ | $4$ | $( 2, 3, 5, 4)( 6,16,26,31)( 7,18,30,34)( 8,20,29,32)( 9,17,28,35)(10,19,27,33) (12,13,15,14)(22,23,25,24)$ | |
| $ 14, 14, 7 $ | $3600$ | $14$ | $( 1,27,31,12,21,17, 6, 2,26,32,11,22,16, 7)( 3,30,33,15,23,20, 8, 5,28,35,13, 25,18,10)( 4,29,34,14,24,19, 9)$ | |
| $ 35 $ | $1440$ | $35$ | $( 1,30,34,13,22,16,10, 4,28,32,11,25,19, 8, 2,26,35,14,23,17, 6, 5,29,33,12, 21,20, 9, 3,27,31,15,24,18, 7)$ | |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1,26,31,11,21,16, 6)( 2,27,32,12,22,17, 7)( 3,28,33,13,23,18, 8) ( 4,29,34,14,24,19, 9)( 5,30,35,15,25,20,10)$ | |
| $ 35 $ | $1440$ | $35$ | $( 1,29,32,15,23,16, 9, 2,30,33,11,24,17,10, 3,26,34,12,25,18, 6, 4,27,35,13, 21,19, 7, 5,28,31,14,22,20, 8)$ | |
| $ 15, 15, 5 $ | $1120$ | $15$ | $( 1,12,18, 4,15,16, 2,13,19, 5,11,17, 3,14,20)( 6,32,28, 9,35,26, 7,33,29,10, 31,27, 8,34,30)(21,22,23,24,25)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,11,16)( 2,12,17)( 3,13,18)( 4,14,19)( 5,15,20)( 6,31,26)( 7,32,27) ( 8,33,28)( 9,34,29)(10,35,30)$ | |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1400$ | $6$ | $( 1,15,16, 5,11,20)( 2,14,17, 4,12,19)( 3,13,18)( 6,35,26,10,31,30) ( 7,34,27, 9,32,29)( 8,33,28)(21,25)(22,24)$ | |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,30,12, 8,16,35, 2,28,11,10,17,33)( 3,26,15, 7,18,31, 5,27,13, 6,20,32) ( 4,29,14, 9,19,34)(21,25,22,23)$ | |
| $ 12, 12, 6, 4, 1 $ | $4200$ | $12$ | $( 1,27,14, 8,16,32, 4,28,11, 7,19,33)( 2,29,13, 6,17,34, 3,26,12, 9,18,31) ( 5,30,15,10,20,35)(21,22,24,23)$ | |
| $ 15, 5, 5, 5, 5 $ | $280$ | $15$ | $( 1, 2, 3, 4, 5)( 6,27,33, 9,30,31, 7,28,34,10,26,32, 8,29,35)(11,12,13,14,15) (16,17,18,19,20)(21,22,23,24,25)$ | |
| $ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 6,26,31)( 7,27,32)( 8,28,33)( 9,29,34)(10,30,35)$ | |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $350$ | $6$ | $( 1, 5)( 2, 4)( 6,30,31,10,26,35)( 7,29,32, 9,27,34)( 8,28,33)(11,15)(12,14) (16,20)(17,19)(21,25)(22,24)$ | |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1050$ | $6$ | $( 1,17)( 2,16)( 3,20)( 4,19)( 5,18)( 6,32,26, 7,31,27)( 8,35,28,10,33,30) ( 9,34,29)(11,22)(12,21)(13,25)(14,24)(15,23)$ | |
| $ 15, 10, 10 $ | $840$ | $30$ | $( 1,20, 4,18, 2,16, 5,19, 3,17)( 6,35,29, 8,32,26,10,34,28, 7,31,30, 9,33,27) (11,25,14,23,12,21,15,24,13,22)$ | |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29) (10,35,30)(11,21)(12,22)(13,23)(14,24)(15,25)$ | |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,23,17,15)( 2,25,16,13)( 3,22,20,11)( 4,24,19,14)( 5,21,18,12) ( 6,28,32,10,26,33, 7,30,31, 8,27,35)( 9,29,34)$ | |
| $ 12, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,24,18,15)( 2,22,17,12)( 3,25,16,14)( 4,23,20,11)( 5,21,19,13) ( 6,29,33,10,26,34, 8,30,31, 9,28,35)( 7,27,32)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(21,31)(22,34,25,33) (23,32,24,35)(27,29,30,28)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $105$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(21,31)(22,33,25,34) (23,35,24,32)(27,28,30,29)$ | |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1, 6,26)( 2, 9,30, 3, 7,29, 5, 8,27, 4,10,28)(12,14,15,13)(17,19,20,18) (21,31)(22,34,25,33)(23,32,24,35)$ | |
| $ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ | $2100$ | $12$ | $( 1, 6,26)( 2, 8,30, 4, 7,28, 5, 9,27, 3,10,29)(12,13,15,14)(17,18,20,19) (21,31)(22,33,25,34)(23,35,24,32)$ | |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,26,16,31,21)( 2,27,17,32,22)( 3,28,18,33,23)( 4,29,19,34,24) ( 5,30,20,35,25)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2016$ | $5$ | $( 1,27,18,34,25)( 2,28,19,35,21)( 3,29,20,31,22)( 4,30,16,32,23) ( 5,26,17,33,24)( 6, 7, 8, 9,10)(11,12,13,14,15)$ | |
| $ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ | $2520$ | $10$ | $( 1,26,16,31,21)( 2,30,17,35,22, 5,27,20,32,25)( 3,29,18,34,23, 4,28,19,33,24) ( 7,10)( 8, 9)(12,15)(13,14)$ | |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1,34,30,22,16, 4,35,27,21,19, 5,32,26,24,20, 2,31,29,25,17)( 3,33,28,23,18) ( 6,14,10,12)( 7,11, 9,15)( 8,13)$ | |
| $ 20, 5, 4, 4, 2 $ | $2520$ | $20$ | $( 1,33,29,22,16, 3,34,27,21,18, 4,32,26,23,19, 2,31,28,24,17)( 5,35,30,25,20) ( 6,13, 9,12)( 7,11, 8,14)(10,15)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | no | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 50400.h | magma: IdentifyGroup(G);
| |
| Character table: | 39 x 39 character table |
magma: CharacterTable(G);