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Magma
magma: G := TransitiveGroup(35, 50);
Group invariants
| Abstract group: | $A_7:F_5$ | magma: IdentifyGroup(G);
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| 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 | 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 $t$: | $50$ | 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: | $(1,35,22,8,16,5,32,23,6,20,2,33,21,10,17,3,31,25,7,18)(4,34,24,9,19)(11,30,12,28)(13,26,15,27)(14,29)$, $(1,8,20,2,9,16,3,10,17,4,6,18,5,7,19)(11,23,15,22,14,21,13,25,12,24)(26,33,30,32,29,31,28,35,27,34)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ $5040$: $S_7$ $10080$: 28T360 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
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 | Index | Representative |
| 1A | $1^{35}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{14},1^{7}$ | $5$ | $2$ | $14$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)$ |
| 2B | $2^{10},1^{15}$ | $105$ | $2$ | $10$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| 2C | $2^{16},1^{3}$ | $525$ | $2$ | $16$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(12,15)(13,14)(16,31)(17,35)(18,34)(19,33)(20,32)(27,30)(28,29)$ |
| 3A | $3^{5},1^{20}$ | $70$ | $3$ | $10$ | $( 1,11,26)( 2,12,27)( 3,13,28)( 4,14,29)( 5,15,30)$ |
| 3B | $3^{10},1^{5}$ | $280$ | $3$ | $20$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)$ |
| 4A1 | $4^{7},2,1^{5}$ | $105$ | $4$ | $22$ | $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)$ |
| 4A-1 | $4^{7},2,1^{5}$ | $105$ | $4$ | $22$ | $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28)$ |
| 4B1 | $4^{7},2^{3},1$ | $525$ | $4$ | $24$ | $( 1,18, 4,17)( 2,16, 3,19)( 5,20)( 6,28, 9,27)( 7,26, 8,29)(10,30)(11,23,14,22)(12,21,13,24)(15,25)(31,33,34,32)$ |
| 4B-1 | $4^{7},2^{3},1$ | $525$ | $4$ | $24$ | $( 1,17, 4,18)( 2,19, 3,16)( 5,20)( 6,27, 9,28)( 7,29, 8,26)(10,30)(11,22,14,23)(12,24,13,21)(15,25)(31,32,34,33)$ |
| 4C | $4^{5},2^{5},1^{5}$ | $630$ | $4$ | $20$ | $( 1,16, 6,11)( 2,17, 7,12)( 3,18, 8,13)( 4,19, 9,14)( 5,20,10,15)(21,26)(22,27)(23,28)(24,29)(25,30)$ |
| 4D1 | $4^{8},1^{3}$ | $1050$ | $4$ | $24$ | $( 2, 4, 5, 3)( 6,16,21,31)( 7,19,25,33)( 8,17,24,35)( 9,20,23,32)(10,18,22,34)(12,14,15,13)(27,29,30,28)$ |
| 4D-1 | $4^{8},1^{3}$ | $1050$ | $4$ | $24$ | $( 2, 3, 5, 4)( 6,31,21,16)( 7,33,25,19)( 8,35,24,17)( 9,32,23,20)(10,34,22,18)(12,13,15,14)(27,28,30,29)$ |
| 4E | $4^{5},2^{7},1$ | $3150$ | $4$ | $22$ | $( 1,24,11,34)( 2,23,12,33)( 3,22,13,32)( 4,21,14,31)( 5,25,15,35)( 6, 9)( 7, 8)(16,29)(17,28)(18,27)(19,26)(20,30)$ |
| 5A | $5^{7}$ | $4$ | $5$ | $28$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
| 5B | $5^{5},1^{10}$ | $504$ | $5$ | $20$ | $( 1, 6,16,11,31)( 2, 7,17,12,32)( 3, 8,18,13,33)( 4, 9,19,14,34)( 5,10,20,15,35)$ |
| 5C | $5^{7}$ | $2016$ | $5$ | $28$ | $( 1, 9,27,25,13)( 2,10,28,21,14)( 3, 6,29,22,15)( 4, 7,30,23,11)( 5, 8,26,24,12)(16,19,17,20,18)(31,34,32,35,33)$ |
| 6A | $3^{5},2^{10}$ | $210$ | $6$ | $20$ | $( 1,21, 6)( 2,22, 7)( 3,23, 8)( 4,24, 9)( 5,25,10)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)$ |
| 6B | $6^{2},3,2^{8},1^{4}$ | $350$ | $6$ | $20$ | $( 1,21,16)( 2,25,17, 5,22,20)( 3,24,18, 4,23,19)( 7,10)( 8, 9)(12,15)(13,14)(27,30)(28,29)(32,35)(33,34)$ |
| 6C | $6^{2},3,2^{10}$ | $1050$ | $6$ | $22$ | $( 1,26,11)( 2,30,12, 5,27,15)( 3,29,13, 4,28,14)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(16,31)(17,35)(18,34)(19,33)(20,32)$ |
| 6D | $6^{4},3^{2},2^{2},1$ | $1400$ | $6$ | $26$ | $( 1, 9,11, 4, 6,14)( 2, 8,12, 3, 7,13)( 5,10,15)(16,29,21,19,26,24)(17,28,22,18,27,23)(20,30,25)(31,34)(32,33)$ |
| 7A | $7^{5}$ | $720$ | $7$ | $30$ | $( 1,31,11,21, 6,16,26)( 2,32,12,22, 7,17,27)( 3,33,13,23, 8,18,28)( 4,34,14,24, 9,19,29)( 5,35,15,25,10,20,30)$ |
| 10A | $10^{2},5^{3}$ | $420$ | $10$ | $30$ | $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| 10B | $10^{2},5,2^{4},1^{2}$ | $2520$ | $10$ | $26$ | $( 1,13, 6,33,16, 3,11, 8,31,18)( 2,12, 7,32,17)( 4,15, 9,35,19, 5,14,10,34,20)(21,23)(24,25)(26,28)(29,30)$ |
| 12A1 | $12,4^{5},3$ | $2100$ | $12$ | $28$ | $( 1,11,26)( 2,13,30, 4,12,28, 5,14,27, 3,15,29)( 6,31,21,16)( 7,33,25,19)( 8,35,24,17)( 9,32,23,20)(10,34,22,18)$ |
| 12A-1 | $12,4^{5},3$ | $2100$ | $12$ | $28$ | $( 1,26,11)( 2,29,15, 3,27,14, 5,28,12, 4,30,13)( 6,16,21,31)( 7,19,25,33)( 8,17,24,35)( 9,20,23,32)(10,18,22,34)$ |
| 12B1 | $12,4^{4},3,2,1^{2}$ | $2100$ | $12$ | $26$ | $( 1,16,21)( 2,19,25, 3,17,24, 5,18,22, 4,20,23)( 6,31)( 7,34,10,33)( 8,32, 9,35)(12,14,15,13)(27,29,30,28)$ |
| 12B-1 | $12,4^{4},3,2,1^{2}$ | $2100$ | $12$ | $26$ | $( 1,21,16)( 2,23,20, 4,22,18, 5,24,17, 3,25,19)( 6,31)( 7,33,10,34)( 8,35, 9,32)(12,13,15,14)(27,28,30,29)$ |
| 12C1 | $12^{2},6,4,1$ | $4200$ | $12$ | $30$ | $( 1,22, 9,18,11,27, 4,23, 6,17,14,28)( 2,24, 8,16,12,29, 3,21, 7,19,13,26)( 5,25,10,20,15,30)(31,32,34,33)$ |
| 12C-1 | $12^{2},6,4,1$ | $4200$ | $12$ | $30$ | $( 1,28,14,17, 6,23, 4,27,11,18, 9,22)( 2,26,13,19, 7,21, 3,29,12,16, 8,24)( 5,30,15,20,10,25)(31,33,34,32)$ |
| 14A | $14^{2},7$ | $3600$ | $14$ | $32$ | $( 1, 8,31,18,11,28,21, 3, 6,33,16,13,26,23)( 2, 7,32,17,12,27,22)( 4,10,34,20,14,30,24, 5, 9,35,19,15,29,25)$ |
| 15A | $15,5^{4}$ | $280$ | $15$ | $30$ | $( 1,23,10, 2,24, 6, 3,25, 7, 4,21, 8, 5,22, 9)(11,13,15,12,14)(16,18,20,17,19)(26,28,30,27,29)(31,33,35,32,34)$ |
| 15B | $15^{2},5$ | $1120$ | $15$ | $32$ | $( 1, 2, 3, 4, 5)( 6,12,33, 9,15,31, 7,13,34,10,11,32, 8,14,35)(16,22,28,19,25,26,17,23,29,20,21,27,18,24,30)$ |
| 20A | $20,10,5$ | $2520$ | $20$ | $32$ | $( 1,19, 7,15, 3,16, 9,12, 5,18, 6,14, 2,20, 8,11, 4,17,10,13)(21,29,22,30,23,26,24,27,25,28)(31,34,32,35,33)$ |
| 20B1 | $20,5,4^{2},2$ | $2520$ | $20$ | $30$ | $( 1,34,13,20, 6, 4,33,15,16, 9, 3,35,11,19, 8, 5,31,14,18,10)( 2,32,12,17, 7)(21,29,23,30)(22,27)(24,28,25,26)$ |
| 20B-1 | $20,5,4^{2},2$ | $2520$ | $20$ | $30$ | $( 1,10,18,14,31, 5, 8,19,11,35, 3, 9,16,15,33, 4, 6,20,13,34)( 2, 7,17,12,32)(21,30,23,29)(22,27)(24,26,25,28)$ |
| 30A | $15,10^{2}$ | $840$ | $30$ | $32$ | $( 1, 7,23, 4,10,21, 2, 8,24, 5, 6,22, 3, 9,25)(11,27,13,29,15,26,12,28,14,30)(16,32,18,34,20,31,17,33,19,35)$ |
| 35A1 | $35$ | $1440$ | $35$ | $34$ | $( 1,23,15,17, 9,26,33, 5,22,14,16, 8,30,32, 4,21,13,20, 7,29,31, 3,25,12,19, 6,28,35, 2,24,11,18,10,27,34)$ |
| 35A-1 | $35$ | $1440$ | $35$ | $34$ | $( 1,34,27,10,18,11,24, 2,35,28, 6,19,12,25, 3,31,29, 7,20,13,21, 4,32,30, 8,16,14,22, 5,33,26, 9,17,15,23)$ |
Malle's constant $a(G)$: $1/10$
magma: ConjugacyClasses(G);
Character table
39 x 39 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed