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Magma
magma: G := TransitiveGroup(35, 10);
Group invariants
| Abstract group: | $C_5\times F_7$ | magma: IdentifyGroup(G);
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| Order: | $210=2 \cdot 3 \cdot 5 \cdot 7$ | 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: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,28,25,2,29,21,3,30,22,4,26,23,5,27,24)(6,13,35,7,14,31,8,15,32,9,11,33,10,12,34)(16,18,20,17,19)$, $(1,16,21,11,31,26)(2,17,22,12,32,27)(3,18,23,13,33,28)(4,19,24,14,34,29)(5,20,25,15,35,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $5$: $C_5$ $6$: $C_6$ $10$: $C_{10}$ $15$: $C_{15}$ $30$: $C_{30}$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
Degree 7: $F_7$
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 | Index | Representative |
| 1A | $1^{35}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{15},1^{5}$ | $7$ | $2$ | $15$ | $( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21)(17,22)(18,23)(19,24)(20,25)$ |
| 3A1 | $3^{10},1^{5}$ | $7$ | $3$ | $20$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32)(18,28,33)(19,29,34)(20,30,35)$ |
| 3A-1 | $3^{10},1^{5}$ | $7$ | $3$ | $20$ | $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27)(18,33,28)(19,34,29)(20,35,30)$ |
| 5A1 | $5^{7}$ | $1$ | $5$ | $28$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$ |
| 5A-1 | $5^{7}$ | $1$ | $5$ | $28$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| 5A2 | $5^{7}$ | $1$ | $5$ | $28$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ |
| 5A-2 | $5^{7}$ | $1$ | $5$ | $28$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
| 6A1 | $6^{5},1^{5}$ | $7$ | $6$ | $25$ | $( 6,26,21,31,11,16)( 7,27,22,32,12,17)( 8,28,23,33,13,18)( 9,29,24,34,14,19)(10,30,25,35,15,20)$ |
| 6A-1 | $6^{5},1^{5}$ | $7$ | $6$ | $25$ | $( 6,16,11,31,21,26)( 7,17,12,32,22,27)( 8,18,13,33,23,28)( 9,19,14,34,24,29)(10,20,15,35,25,30)$ |
| 7A | $7^{5}$ | $6$ | $7$ | $30$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ |
| 10A1 | $10^{3},5$ | $7$ | $10$ | $31$ | $( 1,25, 4,23, 2,21, 5,24, 3,22)( 6,20, 9,18, 7,16,10,19, 8,17)(11,15,14,13,12)(26,35,29,33,27,31,30,34,28,32)$ |
| 10A-1 | $10^{3},5$ | $7$ | $10$ | $31$ | $( 1,13, 5,12, 4,11, 3,15, 2,14)( 6, 8,10, 7, 9)(16,33,20,32,19,31,18,35,17,34)(21,28,25,27,24,26,23,30,22,29)$ |
| 10A3 | $10^{3},5$ | $7$ | $10$ | $31$ | $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,32,13,34,15,31,12,33,14,35)(16,27,18,29,20,26,17,28,19,30)(21,22,23,24,25)$ |
| 10A-3 | $10^{3},5$ | $7$ | $10$ | $31$ | $( 1,19, 2,20, 3,16, 4,17, 5,18)( 6,14, 7,15, 8,11, 9,12,10,13)(21,34,22,35,23,31,24,32,25,33)(26,29,27,30,28)$ |
| 15A1 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,18,15, 2,19,11, 3,20,12, 4,16,13, 5,17,14)( 6,28,35, 7,29,31, 8,30,32, 9,26,33,10,27,34)(21,23,25,22,24)$ |
| 15A-1 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,17, 8, 4,20, 6, 2,18, 9, 5,16, 7, 3,19,10)(11,22,28,14,25,26,12,23,29,15,21,27,13,24,30)(31,32,33,34,35)$ |
| 15A2 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,14,17, 5,13,16, 4,12,20, 3,11,19, 2,15,18)( 6,34,27,10,33,26, 9,32,30, 8,31,29, 7,35,28)(21,24,22,25,23)$ |
| 15A-2 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,27, 8, 4,30, 6, 2,28, 9, 5,26, 7, 3,29,10)(11,12,13,14,15)(16,22,33,19,25,31,17,23,34,20,21,32,18,24,35)$ |
| 15A4 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,30,24, 3,27,21, 5,29,23, 2,26,25, 4,28,22)( 6,15,34, 8,12,31,10,14,33, 7,11,35, 9,13,32)(16,20,19,18,17)$ |
| 15A-4 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,33,15, 2,34,11, 3,35,12, 4,31,13, 5,32,14)( 6,18,25, 7,19,21, 8,20,22, 9,16,23,10,17,24)(26,28,30,27,29)$ |
| 15A7 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1, 9,17, 5, 8,16, 4, 7,20, 3, 6,19, 2,10,18)(11,29,22,15,28,21,14,27,25,13,26,24,12,30,23)(31,34,32,35,33)$ |
| 15A-7 | $15^{2},5$ | $7$ | $15$ | $32$ | $( 1,35,24, 3,32,21, 5,34,23, 2,31,25, 4,33,22)( 6,10, 9, 8, 7)(11,20,29,13,17,26,15,19,28,12,16,30,14,18,27)$ |
| 30A1 | $30,5$ | $7$ | $30$ | $33$ | $( 1,34, 7,25,28,16, 4,32,10,23,26,19, 2,35, 8,21,29,17, 5,33, 6,24,27,20, 3,31, 9,22,30,18)(11,14,12,15,13)$ |
| 30A-1 | $30,5$ | $7$ | $30$ | $33$ | $( 1,23,20,27, 9,11, 3,25,17,29, 6,13, 5,22,19,26, 8,15, 2,24,16,28,10,12, 4,21,18,30, 7,14)(31,33,35,32,34)$ |
| 30A7 | $30,5$ | $7$ | $30$ | $33$ | $( 1,29,32,15,23,16, 4,27,35,13,21,19, 2,30,33,11,24,17, 5,28,31,14,22,20, 3,26,34,12,25,18)( 6, 9, 7,10, 8)$ |
| 30A-7 | $30,5$ | $7$ | $30$ | $33$ | $( 1,22,13,19,35, 6, 2,23,14,20,31, 7, 3,24,15,16,32, 8, 4,25,11,17,33, 9, 5,21,12,18,34,10)(26,27,28,29,30)$ |
| 30A11 | $30,5$ | $7$ | $30$ | $33$ | $( 1,12,28,34,25, 6, 2,13,29,35,21, 7, 3,14,30,31,22, 8, 4,15,26,32,23, 9, 5,11,27,33,24,10)(16,17,18,19,20)$ |
| 30A-11 | $30,5$ | $7$ | $30$ | $33$ | $( 1,10,34,18,12,21, 5, 9,33,17,11,25, 4, 8,32,16,15,24, 3, 7,31,20,14,23, 2, 6,35,19,13,22)(26,30,29,28,27)$ |
| 30A13 | $30,5$ | $7$ | $30$ | $33$ | $( 1, 8,25,32,29,11, 3,10,22,34,26,13, 5, 7,24,31,28,15, 2, 9,21,33,30,12, 4, 6,23,35,27,14)(16,18,20,17,19)$ |
| 30A-13 | $30,5$ | $7$ | $30$ | $33$ | $( 1,15, 9,28,17,21, 5,14, 8,27,16,25, 4,13, 7,26,20,24, 3,12, 6,30,19,23, 2,11,10,29,18,22)(31,35,34,33,32)$ |
| 35A1 | $35$ | $6$ | $35$ | $34$ | $( 1,32,28,24,20,11, 7, 3,34,30,21,17,13, 9, 5,31,27,23,19,15, 6, 2,33,29,25,16,12, 8, 4,35,26,22,18,14,10)$ |
| 35A-1 | $35$ | $6$ | $35$ | $34$ | $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12,26,10,24, 3,17,31,15,29, 8,22)$ |
| 35A2 | $35$ | $6$ | $35$ | $34$ | $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9,31,23,15, 2,29,16, 8,35,22,14)$ |
| 35A-2 | $35$ | $6$ | $35$ | $34$ | $( 1,24, 7,30,13,31,19, 2,25, 8,26,14,32,20, 3,21, 9,27,15,33,16, 4,22,10,28,11,34,17, 5,23, 6,29,12,35,18)$ |
Malle's constant $a(G)$: $1/15$
magma: ConjugacyClasses(G);
Character table
35 x 35 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed