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Magma
magma: G := TransitiveGroup(35, 22);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7:\GL(2,4)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | 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: | (1,15,24,34,7,19,28,2,13,21,35,6,20,30)(3,11,25,33,8,17,29,4,14,23,31,9,16,27)(5,12,22,32,10,18,26), (1,28,24)(2,26,25)(3,30,22)(4,27,23)(5,29,21)(6,12,31)(7,13,35)(8,15,32)(9,11,33)(10,14,34)(16,19,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $21$: $C_7:C_3$ $60$: $A_5$ $180$: $\GL(2,4)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $A_5$
Degree 7: $C_7:C_3$
Low degree siblings
42T197Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,20,35,13,28, 7,24)( 2,19,34,15,30, 6,21)( 3,16,31,14,29, 8,25) ( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$ |
| $ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,30,34)(20,28,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,14)(13,15)(16,17)(19,20)(21,24)(23,25)(27,29) (28,30)(31,33)(34,35)$ |
| $ 14, 14, 7 $ | $45$ | $14$ | $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,17,31,11,29, 9,25, 4,16,33,14, 27, 8,23)( 5,18,32,12,26,10,22)$ |
| $ 14, 14, 7 $ | $45$ | $14$ | $( 1,15,24,34, 7,19,28, 2,13,21,35, 6,20,30)( 3,11,25,33, 8,17,29, 4,14,23,31, 9,16,27)( 5,12,22,32,10,18,26)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1, 2)( 3, 4)( 6,24,15, 7,21,13)( 8,23,14, 9,25,11)(10,22,12)(16,27,31,17,29, 33)(18,26,32)(19,28,34,20,30,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1, 2)( 3, 4)( 6,13,21, 7,15,24)( 8,11,25, 9,14,23)(10,12,22)(16,33,29,17,31, 27)(18,32,26)(19,35,30,20,34,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 2, 3)( 6, 8, 7)(13,15,14)(16,20,19)(21,25,24)(28,30,29)(31,35,34)$ |
| $ 21, 7, 7 $ | $60$ | $21$ | $( 1,19,31,13,30, 8,24, 2,16,35,15,29, 7,21, 3,20,34,14,28, 6,25) ( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$ |
| $ 21, 7, 7 $ | $60$ | $21$ | $( 1,15,25,35, 6,16,28, 2,14,24,34, 8,20,30, 3,13,21,31, 7,19,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1, 2, 3)( 6,25,13)( 7,21,14)( 8,24,15)( 9,23,11)(10,22,12)(16,28,34) (17,27,33)(18,26,32)(19,29,35)(20,30,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1, 2, 3)( 6,14,24)( 7,15,25)( 8,13,21)( 9,11,23)(10,12,22)(16,35,30) (17,33,27)(18,32,26)(19,31,28)(20,34,29)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 8, 9,10, 7)(11,12,13,15,14)(16,17,18,20,19) (21,25,23,22,24)(26,28,30,29,27)(31,33,32,35,34)$ |
| $ 35 $ | $36$ | $35$ | $( 1,19,31,11,26, 7,21, 3,17,32,13,30, 8,23, 5,20,34,14,27,10,24, 2,16,33,12, 28, 6,25, 4,18,35,15,29, 9,22)$ |
| $ 35 $ | $36$ | $35$ | $( 1,15,25,33,10,20,30, 3,11,22,35, 6,16,27, 5,13,21,31, 9,18,28, 2,14,23,32, 7,19,29, 4,12,24,34, 8,17,26)$ |
| $ 15, 15, 5 $ | $84$ | $15$ | $( 1, 2, 3, 4, 5)( 6,25,11,10,24,15, 8,23,12, 7,21,14, 9,22,13)(16,27,32,20,30, 31,17,26,35,19,29,33,18,28,34)$ |
| $ 15, 15, 5 $ | $84$ | $15$ | $( 1, 2, 3, 4, 5)( 6,14,23,10,13,21, 8,11,22, 7,15,25, 9,12,24)(16,33,26,20,34, 29,17,32,28,19,31,27,18,35,30)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 8,10, 9, 7)(11,13,15,14,12)(16,18,17,20,19) (21,25,22,23,24)(26,27,28,30,29)(31,32,33,35,34)$ |
| $ 35 $ | $36$ | $35$ | $( 1,19,31,12,27, 7,21, 3,18,33,13,30, 8,22, 4,20,34,14,26, 9,24, 2,16,32,11, 28, 6,25, 5,17,35,15,29,10,23)$ |
| $ 35 $ | $36$ | $35$ | $( 1,15,25,32, 9,20,30, 3,12,23,35, 6,16,26, 4,13,21,31,10,17,28, 2,14,22,33, 7,19,29, 5,11,24,34, 8,18,27)$ |
| $ 15, 15, 5 $ | $84$ | $15$ | $( 1, 2, 3, 5, 4)( 6,25,12, 9,24,15, 8,22,11, 7,21,14,10,23,13)(16,26,33,20,30, 31,18,27,35,19,29,32,17,28,34)$ |
| $ 15, 15, 5 $ | $84$ | $15$ | $( 1, 2, 3, 5, 4)( 6,14,22, 9,13,21, 8,12,23, 7,15,25,10,11,24)(16,32,27,20,34, 29,18,33,28,19,31,26,17,35,30)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1260=2^{2} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 1260.61 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);