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Magma
magma: G := TransitiveGroup(35, 42);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5:S_7$ | ||
| 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: | (1,18,26,23,31,3,16,28,21,33)(2,17,27,22,32)(4,20,29,25,34,5,19,30,24,35)(6,13)(7,12)(8,11)(9,15)(10,14), (1,29,31,14,6,4,26,34,11,9)(2,28,32,13,7,3,27,33,12,8)(5,30,35,15,10)(16,24)(17,23)(18,22)(19,21)(20,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ $5040$: $S_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
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 $ | $2$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1, 3, 5, 2, 4)( 6,28,10,27, 9,26, 8,30, 7,29)(11,13,15,12,14) (16,18,20,17,19)(21,33,25,32,24,31,23,35,22,34)$ | |
| $ 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 $ | $105$ | $2$ | $( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(21,31)(22,32)(23,33)(24,34)(25,35)$ | |
| $ 10, 10, 5, 5, 5 $ | $210$ | $10$ | $( 1, 5, 4, 3, 2)( 6,30, 9,28, 7,26,10,29, 8,27)(11,15,14,13,12) (16,20,19,18,17)(21,35,24,33,22,31,25,34,23,32)$ | |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1, 3, 5, 2, 4)( 6,23,30,32, 9,21,28,35, 7,24,26,33,10,22,29,31, 8,25,27,34) (11,18,15,17,14,16,13,20,12,19)$ | |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $630$ | $4$ | $( 6,21,26,31)( 7,22,27,32)( 8,23,28,33)( 9,24,29,34)(10,25,30,35)(11,16) (12,17)(13,18)(14,19)(15,20)$ | |
| $ 20, 10, 5 $ | $1260$ | $20$ | $( 1, 5, 4, 3, 2)( 6,25,29,33, 7,21,30,34, 8,22,26,35, 9,23,27,31,10,24,28,32) (11,20,14,18,12,16,15,19,13,17)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $525$ | $2$ | $( 1, 5)( 2, 4)( 6,30)( 7,29)( 8,28)( 9,27)(10,26)(11,20)(12,19)(13,18)(14,17) (15,16)(21,35)(22,34)(23,33)(24,32)(25,31)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(21,31)(22,35)(23,34) (24,33)(25,32)(27,30)(28,29)$ | |
| $ 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 $ | $140$ | $15$ | $( 1, 9,12, 5, 8,11, 4, 7,15, 3, 6,14, 2,10,13)(16,19,17,20,18)(21,24,22,25,23) (26,29,27,30,28)(31,34,32,35,33)$ | |
| $ 15, 5, 5, 5, 5 $ | $140$ | $15$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25) (26,27,28,29,30)(31,32,33,34,35)$ | |
| $ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $2100$ | $6$ | $( 1,13, 6, 3,11, 8)( 2,12, 7)( 4,15, 9, 5,14,10)(16,18)(19,20)(21,33)(22,32) (23,31)(24,35)(25,34)(26,28)(29,30)$ | |
| $ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)(16,26)(17,27)(18,28)(19,29) (20,30)(21,31)(22,32)(23,33)(24,34)(25,35)$ | |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1, 9,12, 5, 8,11, 4, 7,15, 3, 6,14, 2,10,13)(16,29,17,30,18,26,19,27,20,28) (21,34,22,35,23,31,24,32,25,33)$ | |
| $ 15, 10, 10 $ | $420$ | $30$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,27,18,29,20,26,17,28,19,30) (21,32,23,34,25,31,22,33,24,35)$ | |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $1050$ | $4$ | $( 1, 4)( 2, 3)( 6,34,26,24)( 7,33,27,23)( 8,32,28,22)( 9,31,29,21) (10,35,30,25)(11,14)(12,13)(16,19)(17,18)$ | |
| $ 6, 6, 4, 4, 4, 4, 4, 3 $ | $2100$ | $12$ | $( 1,20,11, 5,16,15)( 2,19,12, 4,17,14)( 3,18,13)( 6,25,26,35)( 7,24,27,34) ( 8,23,28,33)( 9,22,29,32)(10,21,30,31)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,27, 8,34,25)( 2,28, 9,35,21)( 3,29,10,31,22)( 4,30, 6,32,23) ( 5,26, 7,33,24)(11,12,13,14,15)(16,17,18,19,20)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1008$ | $5$ | $( 1,28,10,32,24)( 2,29, 6,33,25)( 3,30, 7,34,21)( 4,26, 8,35,22) ( 5,27, 9,31,23)(11,13,15,12,14)(16,18,20,17,19)$ | |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,26, 6,31,21)( 2,27, 7,32,22)( 3,28, 8,33,23)( 4,29, 9,34,24) ( 5,30,10,35,25)$ | |
| $ 10, 10, 5, 2, 2, 2, 2, 2 $ | $2520$ | $10$ | $( 1,31,26,21, 6)( 2,35,27,25, 7, 5,32,30,22,10)( 3,34,28,24, 8, 4,33,29,23, 9) (11,16)(12,20)(13,19)(14,18)(15,17)$ | |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,29,12, 5,28,11, 4,27,15, 3,26,14, 2,30,13)( 6, 9, 7,10, 8)(16,34,22,20,33, 21,19,32,25,18,31,24,17,35,23)$ | |
| $ 15, 15, 5 $ | $560$ | $15$ | $( 1,27,13, 4,30,11, 2,28,14, 5,26,12, 3,29,15)( 6, 7, 8, 9,10)(16,32,23,19,35, 21,17,33,24,20,31,22,18,34,25)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $280$ | $3$ | $( 1,26,11)( 2,27,12)( 3,28,13)( 4,29,14)( 5,30,15)(16,31,21)(17,32,22) (18,33,23)(19,34,24)(20,35,25)$ | |
| $ 6, 6, 6, 6, 6, 2, 2, 1 $ | $4200$ | $6$ | $( 1,32,26,22,11,17)( 2,31,27,21,12,16)( 3,35,28,25,13,20)( 4,34,29,24,14,19) ( 5,33,30,23,15,18)( 6, 7)( 8,10)$ | |
| $ 35 $ | $720$ | $35$ | $( 1,23,15,32,29, 6,18, 5,22,14,31,28,10,17, 4,21,13,35,27, 9,16, 3,25,12,34, 26, 8,20, 2,24,11,33,30, 7,19)$ | |
| $ 7, 7, 7, 7, 7 $ | $720$ | $7$ | $( 1,21,11,31,26, 6,16)( 2,22,12,32,27, 7,17)( 3,23,13,33,28, 8,18) ( 4,24,14,34,29, 9,19)( 5,25,15,35,30,10,20)$ | |
| $ 35 $ | $720$ | $35$ | $( 1,24,12,35,28, 6,19, 2,25,13,31,29, 7,20, 3,21,14,32,30, 8,16, 4,22,15,33, 26, 9,17, 5,23,11,34,27,10,18)$ | |
| $ 35 $ | $720$ | $35$ | $( 1,25,14,33,27, 6,20, 4,23,12,31,30, 9,18, 2,21,15,34,28, 7,16, 5,24,13,32, 26,10,19, 3,22,11,35,29, 8,17)$ | |
| $ 35 $ | $720$ | $35$ | $( 1,22,13,34,30, 6,17, 3,24,15,31,27, 8,19, 5,21,12,33,29,10,16, 2,23,14,35, 26, 7,18, 4,25,11,32,28, 9,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | 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: | 25200.m | magma: IdentifyGroup(G);
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| Character table: | 33 x 33 character table |
magma: CharacterTable(G);