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Magma
magma: G := TransitiveGroup(35, 48);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $48$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $F_5\times A_7$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | 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: | (1,25,18,9,26,35,13,4,21,20,8,29,31,15,3,24,16,10,28,34,11,5,23,19,6,30,33,14)(2,22,17,7,27,32,12), (1,6)(2,10)(3,9)(4,8)(5,7)(11,16,26,31)(12,20,27,35)(13,19,28,34)(14,18,29,33)(15,17,30,32)(22,25)(23,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ $2520$: $A_7$ $5040$: $A_7\times C_2$ $10080$: 28T362 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $A_7$
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
| 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 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) (32,33,35,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) (32,34,35,33)$ |
| $ 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 $ | $105$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| $ 10, 10, 5, 5, 5 $ | $420$ | $10$ | $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25) (26,27,28,29,30)(31,32,33,34,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $525$ | $2$ | $( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25) (23,24)(27,30)(28,29)(32,35)(33,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $525$ | $4$ | $( 1, 6)( 2, 8, 5, 9)( 3,10, 4, 7)(11,16)(12,18,15,19)(13,20,14,17) (22,23,25,24)(27,28,30,29)(32,33,35,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $525$ | $4$ | $( 1, 6)( 2, 9, 5, 8)( 3, 7, 4,10)(11,16)(12,19,15,18)(13,17,14,20) (22,24,25,23)(27,29,30,28)(32,34,35,33)$ |
| $ 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 $ | $70$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
| $ 15, 5, 5, 5, 5 $ | $280$ | $15$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25) (26,27,28,29,30)(31,32,33,34,35)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $350$ | $6$ | $( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(17,20)(18,19)(22,25)(23,24) (27,30)(28,29)(32,35)(33,34)$ |
| $ 12, 4, 4, 4, 4, 3, 1, 1, 1, 1 $ | $350$ | $12$ | $( 1, 6,11)( 2, 8,15, 4, 7,13, 5, 9,12, 3,10,14)(17,18,20,19)(22,23,25,24) (27,28,30,29)(32,33,35,34)$ |
| $ 12, 4, 4, 4, 4, 3, 1, 1, 1, 1 $ | $350$ | $12$ | $( 1, 6,11)( 2, 9,15, 3, 7,14, 5, 8,12, 4,10,13)(17,19,20,18)(22,24,25,23) (27,29,30,28)(32,34,35,33)$ |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)(16,21)(17,22)(18,23)(19,24) (20,25)(26,31)(27,32)(28,33)(29,34)(30,35)$ |
| $ 15, 10, 10 $ | $840$ | $30$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,22,18,24,20,21,17,23,19,25) (26,32,28,34,30,31,27,33,29,35)$ |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1050$ | $6$ | $( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(16,21)(17,25)(18,24)(19,23) (20,22)(26,31)(27,35)(28,34)(29,33)(30,32)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 2 $ | $1050$ | $12$ | $( 1, 6,11)( 2, 8,15, 4, 7,13, 5, 9,12, 3,10,14)(16,21)(17,23,20,24) (18,25,19,22)(26,31)(27,33,30,34)(28,35,29,32)$ |
| $ 12, 4, 4, 4, 4, 3, 2, 2 $ | $1050$ | $12$ | $( 1, 6,11)( 2, 9,15, 3, 7,14, 5, 8,12, 4,10,13)(16,21)(17,24,20,23) (18,22,19,25)(26,31)(27,34,30,33)(28,32,29,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)(16,21,26)(17,22,27) (18,23,28)(19,24,29)(20,25,30)$ |
| $ 15, 15, 5 $ | $1120$ | $15$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,22,28,19,25,26,17,23,29,20, 21,27,18,24,30)(31,32,33,34,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1400$ | $6$ | $( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(16,21,26)(17,25,27,20,22,30) (18,24,28,19,23,29)(32,35)(33,34)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $1400$ | $12$ | $( 1, 6,11)( 2, 8,15, 4, 7,13, 5, 9,12, 3,10,14)(16,21,26)(17,23,30,19,22,28, 20,24,27,18,25,29)(32,33,35,34)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $1400$ | $12$ | $( 1, 6,11)( 2, 9,15, 3, 7,14, 5, 8,12, 4,10,13)(16,21,26)(17,24,30,18,22,29, 20,23,27,19,25,28)(32,34,35,33)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 1, 6,11,16)( 2, 7,12,17)( 3, 8,13,18)( 4, 9,14,19)( 5,10,15,20)(21,26) (22,27)(23,28)(24,29)(25,30)$ |
| $ 20, 10, 5 $ | $2520$ | $20$ | $( 1, 7,13,19, 5, 6,12,18, 4,10,11,17, 3, 9,15,16, 2, 8,14,20)(21,27,23,29,25, 26,22,28,24,30)(31,32,33,34,35)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $3150$ | $4$ | $( 1, 6,11,16)( 2,10,12,20)( 3, 9,13,19)( 4, 8,14,18)( 5, 7,15,17)(21,26) (22,30)(23,29)(24,28)(25,27)(32,35)(33,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $3150$ | $4$ | $( 1, 6,11,16)( 2, 8,15,19)( 3,10,14,17)( 4, 7,13,20)( 5, 9,12,18)(21,26) (22,28,25,29)(23,30,24,27)(32,33,35,34)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $3150$ | $4$ | $( 1, 6,11,16)( 2, 9,15,18)( 3, 7,14,20)( 4,10,13,17)( 5, 8,12,19)(21,26) (22,29,25,28)(23,27,24,30)(32,34,35,33)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2016$ | $5$ | $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ | $2520$ | $10$ | $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24) (27,30)(28,29)(32,35)(33,34)$ |
| $ 20, 5, 4, 4, 1, 1 $ | $2520$ | $20$ | $( 1, 6,11,16,21)( 2, 8,15,19,22, 3,10,14,17,23, 5, 9,12,18,25, 4, 7,13,20,24) (27,28,30,29)(32,33,35,34)$ |
| $ 20, 5, 4, 4, 1, 1 $ | $2520$ | $20$ | $( 1, 6,11,16,21)( 2, 9,15,18,22, 4,10,13,17,24, 5, 8,12,19,25, 3, 7,14,20,23) (27,29,30,28)(32,34,35,33)$ |
| $ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
| $ 35 $ | $1440$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
| $ 14, 14, 7 $ | $1800$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ |
| $ 28, 7 $ | $1800$ | $28$ | $( 1, 6,11,16,21,26,31)( 2, 8,15,19,22,28,35, 4, 7,13,20,24,27,33, 5, 9,12,18, 25,29,32, 3,10,14,17,23,30,34)$ |
| $ 28, 7 $ | $1800$ | $28$ | $( 1, 6,11,16,21,26,31)( 2, 9,15,18,22,29,35, 3, 7,14,20,23,27,34, 5, 8,12,19, 25,28,32, 4,10,13,17,24,30,33)$ |
| $ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1, 6,11,16,21,31,26)( 2, 7,12,17,22,32,27)( 3, 8,13,18,23,33,28) ( 4, 9,14,19,24,34,29)( 5,10,15,20,25,35,30)$ |
| $ 35 $ | $1440$ | $35$ | $( 1, 7,13,19,25,31,27, 3, 9,15,16,22,33,29, 5, 6,12,18,24,35,26, 2, 8,14,20, 21,32,28, 4,10,11,17,23,34,30)$ |
| $ 14, 14, 7 $ | $1800$ | $14$ | $( 1, 6,11,16,21,31,26)( 2,10,12,20,22,35,27, 5, 7,15,17,25,32,30) ( 3, 9,13,19,23,34,28, 4, 8,14,18,24,33,29)$ |
| $ 28, 7 $ | $1800$ | $28$ | $( 1, 6,11,16,21,31,26)( 2, 8,15,19,22,33,30, 4, 7,13,20,24,32,28, 5, 9,12,18, 25,34,27, 3,10,14,17,23,35,29)$ |
| $ 28, 7 $ | $1800$ | $28$ | $( 1, 6,11,16,21,31,26)( 2, 9,15,18,22,34,30, 3, 7,14,20,23,32,29, 5, 8,12,19, 25,33,27, 4,10,13,17,24,35,28)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | 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: | 50400.f | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);