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Magma
magma: G := TransitiveGroup(35, 20);
Group invariants
| Abstract group: | $F_5\times F_7$ | magma: IdentifyGroup(G);
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| Order: | $840=2^{3} \cdot 3 \cdot 5 \cdot 7$ | 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 $t$: | $20$ | 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,4,5,2)(6,34,10,32)(7,31,9,35)(8,33)(11,29,15,27)(12,26,14,30)(13,28)(16,24,20,22)(17,21,19,25)(18,23)$, $(1,22,13,19,35,6,2,23,14,20,31,7,3,24,15,16,32,8,4,25,11,17,33,9,5,21,12,18,34,10)(26,27,28,29,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $20$: $F_5$ $24$: 24T2 $40$: $F_{5}\times C_2$ $42$: $F_7$ $60$: $F_5\times C_3$ $84$: $F_7 \times C_2$ $120$: 30T26 $168$: 28T26 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
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 | $2^{14},1^{7}$ | $5$ | $2$ | $14$ | $( 1, 4)( 2, 3)( 6, 9)( 7, 8)(11,14)(12,13)(16,19)(17,18)(21,24)(22,23)(26,29)(27,28)(31,34)(32,33)$ |
| 2B | $2^{15},1^{5}$ | $7$ | $2$ | $15$ | $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,31)(22,32)(23,33)(24,34)(25,35)$ |
| 2C | $2^{17},1$ | $35$ | $2$ | $17$ | $( 1,31)( 2,35)( 3,34)( 4,33)( 5,32)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)(11,21)(12,25)(13,24)(14,23)(15,22)(17,20)(18,19)$ |
| 3A1 | $3^{10},1^{5}$ | $7$ | $3$ | $20$ | $( 1,11,31)( 2,12,32)( 3,13,33)( 4,14,34)( 5,15,35)( 6,21,16)( 7,22,17)( 8,23,18)( 9,24,19)(10,25,20)$ |
| 3A-1 | $3^{10},1^{5}$ | $7$ | $3$ | $20$ | $( 1,31,11)( 2,32,12)( 3,33,13)( 4,34,14)( 5,35,15)( 6,16,21)( 7,17,22)( 8,18,23)( 9,19,24)(10,20,25)$ |
| 4A1 | $4^{7},1^{7}$ | $5$ | $4$ | $21$ | $( 1, 3, 4, 2)( 6, 8, 9, 7)(11,13,14,12)(16,18,19,17)(21,23,24,22)(26,28,29,27)(31,33,34,32)$ |
| 4A-1 | $4^{7},1^{7}$ | $5$ | $4$ | $21$ | $( 1, 2, 4, 3)( 6, 7, 9, 8)(11,12,14,13)(16,17,19,18)(21,22,24,23)(26,27,29,28)(31,32,34,33)$ |
| 4B1 | $4^{7},2^{3},1$ | $35$ | $4$ | $24$ | $( 1, 2, 4, 3)( 6,32, 9,33)( 7,34, 8,31)(10,35)(11,27,14,28)(12,29,13,26)(15,30)(16,22,19,23)(17,24,18,21)(20,25)$ |
| 4B-1 | $4^{7},2^{3},1$ | $35$ | $4$ | $24$ | $( 1, 3, 4, 2)( 6,33, 9,32)( 7,31, 8,34)(10,35)(11,28,14,27)(12,26,13,29)(15,30)(16,23,19,22)(17,21,18,24)(20,25)$ |
| 5A | $5^{7}$ | $4$ | $5$ | $28$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$ |
| 6A1 | $6^{5},1^{5}$ | $7$ | $6$ | $25$ | $( 1,21,11,16,31, 6)( 2,22,12,17,32, 7)( 3,23,13,18,33, 8)( 4,24,14,19,34, 9)( 5,25,15,20,35,10)$ |
| 6A-1 | $6^{5},1^{5}$ | $7$ | $6$ | $25$ | $( 1, 6,31,16,11,21)( 2, 7,32,17,12,22)( 3, 8,33,18,13,23)( 4, 9,34,19,14,24)( 5,10,35,20,15,25)$ |
| 6B1 | $6^{4},3^{2},2^{2},1$ | $35$ | $6$ | $26$ | $( 1,19, 6, 4,16, 9)( 2,18, 7, 3,17, 8)( 5,20,10)(11,24,26,14,21,29)(12,23,27,13,22,28)(15,25,30)(31,34)(32,33)$ |
| 6B-1 | $6^{4},3^{2},2^{2},1$ | $35$ | $6$ | $26$ | $( 1, 9,16, 4, 6,19)( 2, 8,17, 3, 7,18)( 5,10,20)(11,29,21,14,26,24)(12,28,22,13,27,23)(15,30,25)(31,34)(32,33)$ |
| 6C1 | $6^{5},2^{2},1$ | $35$ | $6$ | $27$ | $( 1, 6,21,31,26,11)( 2,10,22,35,27,15)( 3, 9,23,34,28,14)( 4, 8,24,33,29,13)( 5, 7,25,32,30,12)(17,20)(18,19)$ |
| 6C-1 | $6^{5},2^{2},1$ | $35$ | $6$ | $27$ | $( 1,11,26,31,21, 6)( 2,15,27,35,22,10)( 3,14,28,34,23, 9)( 4,13,29,33,24, 8)( 5,12,30,32,25, 7)(17,20)(18,19)$ |
| 7A | $7^{5}$ | $6$ | $7$ | $30$ | $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13)( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$ |
| 10A | $10^{3},5$ | $28$ | $10$ | $31$ | $( 1,18, 5,17, 4,16, 3,20, 2,19)( 6,13,10,12, 9,11, 8,15, 7,14)(21,33,25,32,24,31,23,35,22,34)(26,28,30,27,29)$ |
| 12A1 | $12^{2},4,3^{2},1$ | $35$ | $12$ | $29$ | $( 1, 7,19, 3, 6,17, 4, 8,16, 2, 9,18)( 5,10,20)(11,27,24,13,26,22,14,28,21,12,29,23)(15,30,25)(31,32,34,33)$ |
| 12A-1 | $12^{2},4,3^{2},1$ | $35$ | $12$ | $29$ | $( 1,18, 9, 2,16, 8, 4,17, 6, 3,19, 7)( 5,20,10)(11,23,29,12,21,28,14,22,26,13,24,27)(15,25,30)(31,33,34,32)$ |
| 12A5 | $12^{2},4,3^{2},1$ | $35$ | $12$ | $29$ | $( 1,17, 9, 3,16, 7, 4,18, 6, 2,19, 8)( 5,20,10)(11,22,29,13,21,27,14,23,26,12,24,28)(15,25,30)(31,32,34,33)$ |
| 12A-5 | $12^{2},4,3^{2},1$ | $35$ | $12$ | $29$ | $( 1, 8,19, 2, 6,18, 4, 7,16, 3, 9,17)( 5,10,20)(11,28,24,12,26,23,14,27,21,13,29,22)(15,30,25)(31,33,34,32)$ |
| 12B1 | $12^{2},6,4,1$ | $35$ | $12$ | $30$ | $( 1, 3, 4, 2)( 6,28,24,32,11,18, 9,27,21,33,14,17)( 7,26,23,34,12,16, 8,29,22,31,13,19)(10,30,25,35,15,20)$ |
| 12B-1 | $12^{2},6,4,1$ | $35$ | $12$ | $30$ | $( 1, 2, 4, 3)( 6,17,14,33,21,27, 9,18,11,32,24,28)( 7,19,13,31,22,29, 8,16,12,34,23,26)(10,20,15,35,25,30)$ |
| 12B5 | $12^{2},6,4,1$ | $35$ | $12$ | $30$ | $( 1, 3, 4, 2)( 6,18,14,32,21,28, 9,17,11,33,24,27)( 7,16,13,34,22,26, 8,19,12,31,23,29)(10,20,15,35,25,30)$ |
| 12B-5 | $12^{2},6,4,1$ | $35$ | $12$ | $30$ | $( 1, 2, 4, 3)( 6,27,24,33,11,17, 9,28,21,32,14,18)( 7,29,23,31,12,19, 8,26,22,34,13,16)(10,30,25,35,15,20)$ |
| 14A | $14^{2},7$ | $30$ | $14$ | $32$ | $( 1,12,21,32, 6,17,26, 2,11,22,31, 7,16,27)( 3,15,23,35, 8,20,28, 5,13,25,33,10,18,30)( 4,14,24,34, 9,19,29)$ |
| 15A1 | $15^{2},5$ | $28$ | $15$ | $32$ | $( 1,34,12, 5,33,11, 4,32,15, 3,31,14, 2,35,13)( 6,19,22,10,18,21, 9,17,25, 8,16,24, 7,20,23)(26,29,27,30,28)$ |
| 15A-1 | $15^{2},5$ | $28$ | $15$ | $32$ | $( 1,13,35, 2,14,31, 3,15,32, 4,11,33, 5,12,34)( 6,23,20, 7,24,16, 8,25,17, 9,21,18,10,22,19)(26,28,30,27,29)$ |
| 28A1 | $28,7$ | $30$ | $28$ | $33$ | $( 1,10,12,18,21,30,32, 3, 6,15,17,23,26,35, 2, 8,11,20,22,28,31, 5, 7,13,16,25,27,33)( 4, 9,14,19,24,29,34)$ |
| 28A-1 | $28,7$ | $30$ | $28$ | $33$ | $( 1,33,27,25,16,13, 7, 5,31,28,22,20,11, 8, 2,35,26,23,17,15, 6, 3,32,30,21,18,12,10)( 4,34,29,24,19,14, 9)$ |
| 30A1 | $30,5$ | $28$ | $30$ | $33$ | $( 1,10,34,18,12,21, 5, 9,33,17,11,25, 4, 8,32,16,15,24, 3, 7,31,20,14,23, 2, 6,35,19,13,22)(26,30,29,28,27)$ |
| 30A-1 | $30,5$ | $28$ | $30$ | $33$ | $( 1,10,24,33,27,11, 5, 9,23,32,26,15, 4, 8,22,31,30,14, 3, 7,21,35,29,13, 2, 6,25,34,28,12)(16,20,19,18,17)$ |
| 35A | $35$ | $24$ | $35$ | $34$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17,21,30,34, 3, 7,11,20,24,28,32)$ |
Malle's constant $a(G)$: $1/14$
magma: ConjugacyClasses(G);
Character table
35 x 35 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed