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Magma
magma: G := TransitiveGroup(35, 1);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{35}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $35$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,15,24,33,7,16,30,4,13,22,31,10,19,28,2,11,25,34,8,17,26,5,14,23,32,6,20,29,3,12,21,35,9,18,27) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ $7$: $C_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
Degree 7: $C_7$
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 |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ | |
| $35$ | $1$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ | |
| $35$ | $1$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ | |
| $35$ | $1$ | $35$ | $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ | |
| $35$ | $1$ | $35$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ | |
| $35$ | $1$ | $35$ | $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$ | |
| $35$ | $1$ | $35$ | $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$ | |
| $35$ | $1$ | $35$ | $( 1,14,22,35, 8,16,29, 2,15,23,31, 9,17,30, 3,11,24,32,10,18,26, 4,12,25,33, 6,19,27, 5,13,21,34, 7,20,28)$ | |
| $35$ | $1$ | $35$ | $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ | |
| $35$ | $1$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ | |
| $35$ | $1$ | $35$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ | |
| $35$ | $1$ | $35$ | $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13, 26, 9,22, 5,18,31,14,27,10,23)$ | |
| $35$ | $1$ | $35$ | $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12, 26,10,24, 3,17,31,15,29, 8,22)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18) ( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$ | |
| $35$ | $1$ | $35$ | $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30, 11,32,18, 4,25, 6,27,13,34,20)$ | |
| $35$ | $1$ | $35$ | $( 1,23,10,27,14,31,18, 5,22, 9,26,13,35,17, 4,21, 8,30,12,34,16, 3,25, 7,29, 11,33,20, 2,24, 6,28,15,32,19)$ | |
| $35$ | $1$ | $35$ | $( 1,24, 7,30,13,31,19, 2,25, 8,26,14,32,20, 3,21, 9,27,15,33,16, 4,22,10,28, 11,34,17, 5,23, 6,29,12,35,18)$ | |
| $35$ | $1$ | $35$ | $( 1,25, 9,28,12,31,20, 4,23, 7,26,15,34,18, 2,21,10,29,13,32,16, 5,24, 8,27, 11,35,19, 3,22, 6,30,14,33,17)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13) ( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$ | |
| $35$ | $1$ | $35$ | $( 1,27,18, 9,35,21,12, 3,29,20, 6,32,23,14, 5,26,17, 8,34,25,11, 2,28,19,10, 31,22,13, 4,30,16, 7,33,24,15)$ | |
| $35$ | $1$ | $35$ | $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9, 31,23,15, 2,29,16, 8,35,22,14)$ | |
| $35$ | $1$ | $35$ | $( 1,29,17,10,33,21,14, 2,30,18, 6,34,22,15, 3,26,19, 7,35,23,11, 4,27,20, 8, 31,24,12, 5,28,16, 9,32,25,13)$ | |
| $35$ | $1$ | $35$ | $( 1,30,19, 8,32,21,15, 4,28,17, 6,35,24,13, 2,26,20, 9,33,22,11, 5,29,18, 7, 31,25,14, 3,27,16,10,34,23,12)$ | |
| $7^{5}$ | $1$ | $7$ | $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8) ( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$ | |
| $35$ | $1$ | $35$ | $( 1,32,28,24,20,11, 7, 3,34,30,21,17,13, 9, 5,31,27,23,19,15, 6, 2,33,29,25, 16,12, 8, 4,35,26,22,18,14,10)$ | |
| $35$ | $1$ | $35$ | $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24, 16,13,10, 2,34,26,23,20,12, 9)$ | |
| $35$ | $1$ | $35$ | $( 1,34,27,25,18,11, 9, 2,35,28,21,19,12,10, 3,31,29,22,20,13, 6, 4,32,30,23, 16,14, 7, 5,33,26,24,17,15, 8)$ | |
| $35$ | $1$ | $35$ | $( 1,35,29,23,17,11,10, 4,33,27,21,20,14, 8, 2,31,30,24,18,12, 6, 5,34,28,22, 16,15, 9, 3,32,26,25,19,13, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $35=5 \cdot 7$ | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 35.1 | magma: IdentifyGroup(G);
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| Character table: | 35 x 35 character table |
magma: CharacterTable(G);