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Magma
magma: G := TransitiveGroup(35, 40);
Group invariants
| Abstract group: | $D_5\times A_7$ | magma: IdentifyGroup(G);
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| 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 | 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 $t$: | $40$ | 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: | $(1,33,11,3,31,13)(2,32,12)(4,35,14,5,34,15)(6,23)(7,22)(8,21)(9,25)(10,24)(16,28)(17,27)(18,26)(19,30)(20,29)$, $(1,17,31,7)(2,16,32,6)(3,20,33,10)(4,19,34,9)(5,18,35,8)(11,12)(13,15)(21,27)(22,26)(23,30)(24,29)(25,28)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ $2520$: $A_7$ $5040$: $A_7\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $A_7$
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 | $2^{14},1^{7}$ | $5$ | $2$ | $14$ | $( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)(26,27)(28,30)(31,32)(33,35)$ |
| 2B | $2^{10},1^{15}$ | $105$ | $2$ | $10$ | $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)$ |
| 2C | $2^{16},1^{3}$ | $525$ | $2$ | $16$ | $( 1,31)( 2,35)( 3,34)( 4,33)( 5,32)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(12,15)(13,14)(17,20)(18,19)(27,30)(28,29)$ |
| 3A | $3^{5},1^{20}$ | $70$ | $3$ | $10$ | $( 1,31,21)( 2,32,22)( 3,33,23)( 4,34,24)( 5,35,25)$ |
| 3B | $3^{10},1^{5}$ | $280$ | $3$ | $20$ | $( 1,31,21)( 2,32,22)( 3,33,23)( 4,34,24)( 5,35,25)( 6,26,16)( 7,27,17)( 8,28,18)( 9,29,19)(10,30,20)$ |
| 4A | $4^{5},2^{5},1^{5}$ | $630$ | $4$ | $20$ | $( 1,31,21, 6)( 2,32,22, 7)( 3,33,23, 8)( 4,34,24, 9)( 5,35,25,10)(16,26)(17,27)(18,28)(19,29)(20,30)$ |
| 4B | $4^{5},2^{7},1$ | $3150$ | $4$ | $22$ | $( 1,31,21, 6)( 2,35,22,10)( 3,34,23, 9)( 4,33,24, 8)( 5,32,25, 7)(12,15)(13,14)(16,26)(17,30)(18,29)(19,28)(20,27)$ |
| 5A1 | $5^{7}$ | $2$ | $5$ | $28$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
| 5A2 | $5^{7}$ | $2$ | $5$ | $28$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| 5B | $5^{5},1^{10}$ | $504$ | $5$ | $20$ | $( 1,31,21, 6,26)( 2,32,22, 7,27)( 3,33,23, 8,28)( 4,34,24, 9,29)( 5,35,25,10,30)$ |
| 5C1 | $5^{7}$ | $1008$ | $5$ | $28$ | $( 1,33,25, 7,29)( 2,34,21, 8,30)( 3,35,22, 9,26)( 4,31,23,10,27)( 5,32,24, 6,28)(11,13,15,12,14)(16,18,20,17,19)$ |
| 5C2 | $5^{7}$ | $1008$ | $5$ | $28$ | $( 1,35,24, 8,27)( 2,31,25, 9,28)( 3,32,21,10,29)( 4,33,22, 6,30)( 5,34,23, 7,26)(11,15,14,13,12)(16,20,19,18,17)$ |
| 6A | $3^{5},2^{10}$ | $210$ | $6$ | $20$ | $( 1,31,21)( 2,32,22)( 3,33,23)( 4,34,24)( 5,35,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| 6B | $6^{2},3,2^{8},1^{4}$ | $350$ | $6$ | $20$ | $( 1,35,21, 5,31,25)( 2,34,22, 4,32,24)( 3,33,23)( 6,10)( 7, 9)(11,15)(12,14)(16,20)(17,19)(26,30)(27,29)$ |
| 6C | $6^{2},3,2^{10}$ | $1050$ | $6$ | $22$ | $( 1,33,21, 3,31,23)( 2,32,22)( 4,35,24, 5,34,25)( 6,28)( 7,27)( 8,26)( 9,30)(10,29)(11,18)(12,17)(13,16)(14,20)(15,19)$ |
| 6D | $6^{4},3^{2},2^{2},1$ | $1400$ | $6$ | $26$ | $( 1,31,21)( 2,35,22, 5,32,25)( 3,34,23, 4,33,24)( 6,26,16)( 7,30,17,10,27,20)( 8,29,18, 9,28,19)(12,15)(13,14)$ |
| 7A1 | $7^{5}$ | $360$ | $7$ | $30$ | $( 1,21, 6,26,16,11,31)( 2,22, 7,27,17,12,32)( 3,23, 8,28,18,13,33)( 4,24, 9,29,19,14,34)( 5,25,10,30,20,15,35)$ |
| 7A-1 | $7^{5}$ | $360$ | $7$ | $30$ | $( 1,31,21, 6,26,16,11)( 2,32,22, 7,27,17,12)( 3,33,23, 8,28,18,13)( 4,34,24, 9,29,19,14)( 5,35,25,10,30,20,15)$ |
| 10A1 | $10^{2},5^{3}$ | $210$ | $10$ | $30$ | $( 1,32, 3,34, 5,31, 2,33, 4,35)( 6,22, 8,24,10,21, 7,23, 9,25)(11,12,13,14,15)(16,17,18,19,20)(26,27,28,29,30)$ |
| 10A3 | $10^{2},5^{3}$ | $210$ | $10$ | $30$ | $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6,23,10,22, 9,21, 8,25, 7,24)(11,13,15,12,14)(16,18,20,17,19)(26,28,30,27,29)$ |
| 10B | $10^{2},5,2^{4},1^{2}$ | $2520$ | $10$ | $26$ | $( 1,33,21, 8,26, 3,31,23, 6,28)( 2,32,22, 7,27)( 4,35,24,10,29, 5,34,25, 9,30)(11,13)(14,15)(16,18)(19,20)$ |
| 14A1 | $14^{2},7$ | $1800$ | $14$ | $32$ | $( 1,23, 6,28,16,13,31, 3,21, 8,26,18,11,33)( 2,22, 7,27,17,12,32)( 4,25, 9,30,19,15,34, 5,24,10,29,20,14,35)$ |
| 14A-1 | $14^{2},7$ | $1800$ | $14$ | $32$ | $( 1,31,21, 6,26,16,11)( 2,35,22,10,27,20,12, 5,32,25, 7,30,17,15)( 3,34,23, 9,28,19,13, 4,33,24, 8,29,18,14)$ |
| 15A1 | $15,5^{4}$ | $140$ | $15$ | $30$ | $( 1,35,24, 3,32,21, 5,34,23, 2,31,25, 4,33,22)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(26,30,29,28,27)$ |
| 15A2 | $15,5^{4}$ | $140$ | $15$ | $30$ | $( 1,33,25, 2,34,21, 3,35,22, 4,31,23, 5,32,24)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(26,28,30,27,29)$ |
| 15B1 | $15^{2},5$ | $560$ | $15$ | $32$ | $( 1,32,23, 4,35,21, 2,33,24, 5,31,22, 3,34,25)( 6,27,18, 9,30,16, 7,28,19,10,26,17, 8,29,20)(11,12,13,14,15)$ |
| 15B2 | $15^{2},5$ | $560$ | $15$ | $32$ | $( 1,33,25, 2,34,21, 3,35,22, 4,31,23, 5,32,24)( 6,28,20, 7,29,16, 8,30,17, 9,26,18,10,27,19)(11,13,15,12,14)$ |
| 20A1 | $20,10,5$ | $1260$ | $20$ | $32$ | $( 1,32,23, 9, 5,31,22, 8, 4,35,21, 7, 3,34,25, 6, 2,33,24,10)(11,12,13,14,15)(16,27,18,29,20,26,17,28,19,30)$ |
| 20A3 | $20,10,5$ | $1260$ | $20$ | $32$ | $( 1,33,25, 7, 4,31,23,10, 2,34,21, 8, 5,32,24, 6, 3,35,22, 9)(11,13,15,12,14)(16,28,20,27,19,26,18,30,17,29)$ |
| 30A1 | $15,10^{2}$ | $420$ | $30$ | $32$ | $( 1,33,25, 2,34,21, 3,35,22, 4,31,23, 5,32,24)( 6,28,10,27, 9,26, 8,30, 7,29)(11,18,15,17,14,16,13,20,12,19)$ |
| 30A7 | $15,10^{2}$ | $420$ | $30$ | $32$ | $( 1,35,24, 3,32,21, 5,34,23, 2,31,25, 4,33,22)( 6,30, 9,28, 7,26,10,29, 8,27)(11,20,14,18,12,16,15,19,13,17)$ |
| 35A1 | $35$ | $720$ | $35$ | $34$ | $( 1,25, 9,28,17,11,35, 4,23, 7,26,20,14,33, 2,21,10,29,18,12,31, 5,24, 8,27,16,15,34, 3,22, 6,30,19,13,32)$ |
| 35A-1 | $35$ | $720$ | $35$ | $34$ | $( 1,32,23, 9,30,16,12, 3,34,25, 6,27,18,14, 5,31,22, 8,29,20,11, 2,33,24,10,26,17,13, 4,35,21, 7,28,19,15)$ |
| 35A2 | $35$ | $720$ | $35$ | $34$ | $( 1,23,10,27,19,11,33, 5,22, 9,26,18,15,32, 4,21, 8,30,17,14,31, 3,25, 7,29,16,13,35, 2,24, 6,28,20,12,34)$ |
| 35A-2 | $35$ | $720$ | $35$ | $34$ | $( 1,33,25, 7,29,16,13, 5,32,24, 6,28,20,12, 4,31,23,10,27,19,11, 3,35,22, 9,26,18,15, 2,34,21, 8,30,17,14)$ |
Malle's constant $a(G)$: $1/10$
magma: ConjugacyClasses(G);
Character table
36 x 36 character tablemagma: CharacterTable(G);