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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
| Label | 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 $ | $1$ | $1$ | $()$ | |
| $ 7, 7, 7, 7, 7 $ | $5760$ | $7$ | $( 1,20,30,27,21, 6,10)( 2,28,31,19,16,29,14)( 3, 4,35,17,24,13,33) ( 5,26, 7, 8,15,34,32)( 9,18,23,12,11,22,25)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $420$ | $2$ | $( 1,32)( 2,34)( 3,15)( 4, 6)( 7,21)( 8,14)( 9,13)(10,29)(11,27)(16,23)(17,24) (18,25)(19,20)(22,30)(26,31)(28,35)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 1, 8, 7)( 2, 9,25)( 4,30,11)( 5,12,33)( 6,22,27)(10,31,24)(13,18,34) (14,21,32)(16,35,20)(17,29,26)(19,23,28)$ | |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $3360$ | $6$ | $( 1,21, 8,32, 7,14)( 2,18, 9,34,25,13)( 3,15)( 4,27,30, 6,11,22)( 5,33,12) (10,17,31,29,24,26)(16,19,35,23,20,28)$ | |
| $ 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 $ | $28$ | $2$ | $( 2,11)( 3,15)( 4, 9)( 6,13)(16,23)(18,22)(19,20)(25,30)(27,34)(28,35)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 2 $ | $1120$ | $6$ | $( 1, 7, 8)( 2,30, 9,11,25, 4)( 3,15)( 5,33,12)( 6,34,22,13,27,18)(10,24,31) (14,32,21)(16,19,35,23,20,28)(17,26,29)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,32)( 2,27)( 4,13)( 6, 9)( 7,21)( 8,14)(10,29)(11,34)(17,24)(18,30)(22,25) (26,31)$ | |
| $ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ | $3360$ | $6$ | $( 1,14, 7,32, 8,21)( 2, 6,25,27, 9,22)( 4,18,11,13,30,34)( 5,12,33) (10,26,24,29,31,17)(16,35,20)(19,23,28)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $2$ | $( 1,17)( 2,34)( 3,20)( 4, 6)( 7,31)( 8,14)( 9,13)(10,29)(11,27)(15,19)(16,35) (21,26)(23,28)(24,32)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $2520$ | $4$ | $( 1,21,17,26)( 2, 4,34, 6)( 3,16,20,35)( 7,24,31,32)( 8,29,14,10)( 9,27,13,11) (12,33)(15,23,19,28)(18,22)(25,30)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ | $1260$ | $4$ | $( 1,21,17,26)( 2, 9,34,13)( 3,23,20,28)( 4,27, 6,11)( 7,24,31,32)( 8,29,14,10) (12,33)(15,16,19,35)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $( 1, 8, 7)( 2,19, 6)( 4,16,18)( 5,24,26)( 9,23,22)(10,17,12)(11,20,13) (25,28,27)(29,33,31)(30,35,34)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ | $1120$ | $6$ | $( 1, 7, 8)( 2,13,19,11, 6,20)( 3,15)( 4,22,16, 9,18,23)( 5,26,24)(10,12,17) (25,34,28,30,27,35)(29,31,33)$ | |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1680$ | $6$ | $( 1, 8, 7)( 2,34, 6,35,19,30)( 3,15)( 4,23,18, 9,16,22)( 5,29,26,31,24,33) (10,17,12)(11,27,13,28,20,25)(14,32)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $420$ | $4$ | $( 1,22,26,35)( 2,33,10,11)( 3,21,32,15)( 4,27)( 5,34, 8, 9)( 6,29,12,13) ( 7,23,24,30)(16,25)(17,20,19,31)(18,28)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $1344$ | $5$ | $( 1,14, 5,29,32)( 2,13,34,28,15)( 3,11, 6,27,35)( 4,16,30,22,20) ( 7,33,26, 8,10)( 9,23,25,18,19)(12,17,31,21,24)$ | |
| $ 10, 10, 5, 5, 5 $ | $4032$ | $10$ | $( 1,29,14,32, 5)( 2,35,13, 3,34,11,28, 6,15,27)( 4,18,16,19,30, 9,22,23,20,25) ( 7, 8,33,10,26)(12,21,17,24,31)$ | |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ | $1260$ | $4$ | $( 1, 9,25,14)( 2,22,23, 6)( 4,33,17,35)( 5,30,20,29)( 7,19,28,21)( 8,27,32,15) (10,18)(11,12)(13,24)(16,26)$ | |
| $ 8, 8, 8, 4, 4, 2, 1 $ | $5040$ | $8$ | $( 1,35, 9, 4,25,33,14,17)( 2,32,22,15,23, 8, 6,27)( 3,34)( 5,28,30,21,20, 7, 29,19)(10,12,18,11)(13,16,24,26)$ | |
| $ 15, 15, 5 $ | $2688$ | $15$ | $( 1,22,21,24, 2, 3,23,16,19, 6, 7,17,18,26, 5)( 4, 8,34,15,20)( 9,29,30,14,25, 10,27,35,13,31,12,33,32,11,28)$ | |
| $ 12, 12, 6, 4, 1 $ | $3360$ | $12$ | $( 1, 4,24,28, 8,16,26,27, 7,18, 5,25)( 2,20,29,10,19,13,33,17, 6,11,31,12) ( 3,15,21,32)( 9,30,22,34,23,35)$ |
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: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);