Show commands:
Magma
magma: G := TransitiveGroup(35, 31);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_7$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33), (2,5)(3,7)(9,10)(13,14)(16,21)(17,22)(19,26)(27,33)(28,31)(30,32) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Degree 7: None
Low degree siblings
7T7, 14T46, 21T38, 30T565, 42T411, 42T412, 42T413, 42T418Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $2^{14},1^{7}$ | $105$ | $2$ | $( 1,11)( 2,24)( 3,20)( 4,21)( 6,26)( 7,13)( 8,14)( 9,31)(10,29)(12,33)(17,27) (22,25)(23,28)(32,34)$ | |
| $3^{10},1^{5}$ | $70$ | $3$ | $( 1,31,17)( 2, 4,34)( 3,25,23)( 8,33,10)( 9,27,11)(12,29,14)(15,30,16) (18,35,19)(20,22,28)(21,32,24)$ | |
| $4^{7},2^{3},1$ | $210$ | $4$ | $( 1,24,11, 2)( 3, 8,20,14)( 4,31,21, 9)( 6, 7,26,13)(10,28,29,23)(12,25,33,22) (15,19)(16,35)(17,32,27,34)(18,30)$ | |
| $6^{4},3^{2},2^{2},1$ | $210$ | $6$ | $( 1,27,31,11,17, 9)( 2,32, 4,24,34,21)( 3,28,25,20,23,22)( 6,26)( 7,13) ( 8,29,33,14,10,12)(15,16,30)(18,19,35)$ | |
| $12^{2},6,4,1$ | $420$ | $12$ | $( 1, 4,27,24,31,34,11,21,17, 2, 9,32)( 3,12,28, 8,25,29,20,33,23,14,22,10) ( 6,13,26, 7)(15,18,16,19,30,35)$ | |
| $2^{10},1^{15}$ | $21$ | $2$ | $( 8,15)( 9,21)(10,16)(11,24)(12,18)(13,26)(14,19)(27,32)(29,35)(30,33)$ | |
| $5^{7}$ | $504$ | $5$ | $( 1,20, 5,28,34)( 2,22,17,31, 4)( 3,23, 7,25, 6)( 8,11,26,27,35) ( 9,10,33,18,19)(12,14,21,16,30)(13,32,29,15,24)$ | |
| $10^{2},5^{3}$ | $504$ | $10$ | $( 1,28,20,34, 5)( 2,31,22, 4,17)( 3,25,23, 6, 7)( 8,32,11,29,26,15,27,24,35,13 )( 9,12,10,14,33,21,18,16,19,30)$ | |
| $2^{16},1^{3}$ | $105$ | $2$ | $( 1,29)( 2,13)( 3,27)( 4,26)( 5, 6)( 7,34)( 9,19)(10,20)(11,14)(12,24)(15,33) (16,22)(17,23)(18,21)(25,32)(31,35)$ | |
| $3^{11},1^{2}$ | $280$ | $3$ | $( 1, 9,25)( 2,10,33)( 3,12,31)( 4, 7,22)( 5,23,18)( 6,17,21)( 8,30,28) (13,20,15)(16,26,34)(19,32,29)(24,35,27)$ | |
| $6^{5},3,2$ | $840$ | $6$ | $( 1,32, 9,29,25,19)( 2,15,10,13,33,20)( 3,35,12,27,31,24)( 4,16, 7,26,22,34) ( 5,21,23, 6,18,17)( 8,28,30)(11,14)$ | |
| $4^{7},2^{3},1$ | $630$ | $4$ | $( 1,31,29,35)( 2,20,16,26)( 3,25,27,32)( 4,13,10,22)( 5,14,23,18)( 6,11,17,21) ( 7,33,34,15)( 8,30)( 9,12)(19,24)$ | |
| $7^{5}$ | $720$ | $7$ | $( 1,25,16,34,14,26, 9)( 2,32,13, 8,31,18,23)( 3,28,27,29,33,21,22) ( 4,17, 6,10, 5,12, 7)(11,24,30,15,20,19,35)$ | |
| $6^{3},3^{4},2,1^{3}$ | $420$ | $6$ | $( 1,12,17)( 2,13,34,28,30,20)( 3, 9,23)( 4,16,15)( 5,29,32,31,24,14) ( 6,27,35,25,19,11)( 7, 8,10)(26,33)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 5040.w | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);