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Magma
magma: G := TransitiveGroup(35, 44);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_8$ | ||
| 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), (1,3)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35) | 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
8T50, 16T1838, 28T502, 30T1153Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,33)( 2,18)( 3,26)( 5,25)( 9,20)(10,15)(12,34)(13,23)(14,35)(17,30)(21,28) (24,27)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ | $1260$ | $4$ | $( 1,23,33,13)( 2,20,18, 9)( 3,34,26,12)( 4,19)( 5,24,25,27)( 6, 8)( 7,29) (10,14,15,35)(16,31)(17,21,30,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $420$ | $2$ | $( 1, 2)( 3, 5)( 4, 8)( 6, 7)(10,12)(11,22)(13,23)(14,17)(15,34)(16,29)(18,33) (19,31)(21,27)(24,28)(25,26)(30,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 3,22,28)( 4,14,30)( 5,11,24)( 6,33,16)( 7,18,29)( 8,17,35)( 9,20,32) (10,19,34)(12,31,15)(13,25,21)(23,26,27)$ |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $3360$ | $6$ | $( 1, 2)( 3,11,22,24,28, 5)( 4, 7,14,18,30,29)( 6,15,33,12,16,31) ( 8,10,17,19,35,34)( 9,23,20,26,32,27)(13,21,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $( 4,10,16)( 5,11,24)( 6,14,19)( 7,15, 8)( 9,23,21)(12,17,18)(13,20,26) (25,32,27)(29,31,35)(30,34,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $2$ | $( 1, 3)( 2,22)( 4,20)( 5,23)( 6,17)( 9,24)(10,26)(11,21)(12,19)(13,16)(14,18) (25,31)(27,29)(32,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $420$ | $4$ | $( 1,22, 3, 2)( 4,31,20,25)( 5,17,23, 6)( 7,34)( 8,30)( 9,19,24,12) (10,35,26,32)(11,18,21,14)(13,27,16,29)(15,33)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1680$ | $6$ | $( 1, 3)( 2,22)( 4,13,10,20,16,26)( 5, 9,11,23,24,21)( 6,12,14,17,19,18) ( 7, 8,15)(25,29,32,31,27,35)(30,33,34)$ |
| $ 12, 12, 6, 4, 1 $ | $3360$ | $12$ | $( 1, 2, 3,22)( 4,32,13,31,10,27,20,35,16,25,26,29)( 5,14, 9,17,11,19,23,18,24, 6,21,12)( 7,33, 8,34,15,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $28$ | $2$ | $( 2,24)( 3, 8)( 4,21)( 6,26)(10,23)(12,22)(14,20)(25,33)(28,29)(32,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1344$ | $5$ | $( 1,11,15,35,18)( 2,14,32,26,22)( 3,10,21,28,33)( 4,29,25, 8,23) ( 5,13,30,19,17)( 6,12,24,20,34)( 7, 9,16,31,27)$ |
| $ 10, 10, 5, 5, 5 $ | $4032$ | $10$ | $( 1,35,11,18,15)( 2, 6,14,12,32,24,26,20,22,34)( 3,29,10,25,21, 8,28,23,33, 4) ( 5,19,13,17,30)( 7,31, 9,27,16)$ |
| $ 15, 15, 5 $ | $2688$ | $15$ | $( 1, 3,25,11, 2,28,27,33,10,16,29, 8, 7,23,24)( 4,13,34,26,17,21,12,15, 6,20, 5,22,32,18,14)( 9,30,35,19,31)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ | $3360$ | $6$ | $( 1,26,27)( 2, 6,25,31,21,16)( 3, 7,33,29,24,19)( 5,18,28)( 9,15,10,34,12,20) (11,35,32,17,22,13)(14,23,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $2520$ | $4$ | $( 1,18)( 2,26,31,27)( 3, 6, 7,25)( 4, 8)( 5,33,28,19)( 9,34,10,20) (11,22,35,17)(12,15)(13,30,32,14)(16,24,21,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $1260$ | $4$ | $( 2, 5,31,28)( 3,24, 7,29)( 6,21,25,16)( 9,20,10,34)(11,17,35,22)(12,15) (13,14,32,30)(19,26,33,27)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 2 $ | $1120$ | $6$ | $( 1, 5,17)( 2,27,12)( 3, 9,10,28,13,14)( 4,16,29, 6,19, 8)( 7,31)(11,32,22) (15,21,23,35,26,20)(18,24,34)(25,30,33)$ |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ | $1120$ | $6$ | $( 1, 2)( 3,22,28)( 4, 7,34,31, 6,17)( 8,30,29,19,12,16)( 9,13,27) (10,15,33,35,14,18)(20,25,23)(21,26,32)$ |
| $ 8, 8, 8, 4, 4, 2, 1 $ | $5040$ | $8$ | $( 1, 3,23,34,33,26,13,12)( 2,21,20,30,18,28, 9,17)( 4,31,19,16) ( 5,14,24,15,25,35,27,10)( 6,29, 8, 7)(11,32)$ |
| $ 7, 7, 7, 7, 7 $ | $5760$ | $7$ | $( 1, 3, 7,25,30,12,22)( 2, 8,23,31,32,13,18)( 4,34,29,21, 5,27,16) ( 6,33,24, 9,17,28,15)(10,20,35,14,26,11,19)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 40320.a | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);