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Magma
magma: G := TransitiveGroup(35, 32);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_5\times \GL(3,2)$ | ||
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: | (1,26,22)(2,30,24)(3,28,21)(4,29,23)(5,27,25)(6,8,10)(11,20,34)(12,18,31)(13,16,32)(14,17,33)(15,19,35), (1,27,32,20,3,26,31,17,2,30,35,18,4,29,34,19,5,28,33,16)(6,22,7,24,10,23,9,25,8,21)(11,13,14,15,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $60$: $A_5$ $168$: $\GL(3,2)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $A_5$
Degree 7: $\GL(3,2)$
Low degree siblings
35T32, 40T5863, 42T552 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 6,14)( 7,12)( 8,11)( 9,15)(10,13)(16,34)(17,32)(18,31)(19,35)(20,33)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $56$ | $3$ | $( 6,30,14)( 7,27,12)( 8,28,11)( 9,26,15)(10,29,13)(16,23,34)(17,24,32) (18,25,31)(19,22,35)(20,21,33)$ |
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $42$ | $4$ | $( 6,16)( 7,18)( 8,17)( 9,19)(10,20)(11,24,32,28)(12,25,31,27)(13,21,33,29) (14,23,34,30)(15,22,35,26)$ |
$ 7, 7, 7, 7, 7 $ | $24$ | $7$ | $( 1,26,15,35, 9,22,19)( 2,29,13,33,10,21,20)( 3,30,14,34, 6,23,16) ( 4,28,11,32, 8,24,17)( 5,27,12,31, 7,25,18)$ |
$ 7, 7, 7, 7, 7 $ | $24$ | $7$ | $( 1,26,15,22,19, 9,35)( 2,29,13,21,20,10,33)( 3,30,14,23,16, 6,34) ( 4,28,11,24,17, 8,32)( 5,27,12,25,18, 7,31)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 6, 8)( 9,10)(11,14)(13,15)(16,17)(19,20)(21,22)(23,24)(26,29) (28,30)(32,34)(33,35)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $315$ | $2$ | $( 1, 2)( 3, 4)( 6,11)( 7,12)( 8,14)( 9,13)(10,15)(16,32)(17,34)(18,31)(19,33) (20,35)(21,22)(23,24)(26,29)(28,30)$ |
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $840$ | $6$ | $( 1, 2)( 3, 4)( 6,28,14, 8,30,11)( 7,27,12)( 9,29,15,10,26,13)(16,24,34,17,23, 32)(18,25,31)(19,21,35,20,22,33)$ |
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $630$ | $4$ | $( 1, 2)( 3, 4)( 6,17)( 7,18)( 8,16)( 9,20)(10,19)(11,23,32,30)(12,25,31,27) (13,22,33,26)(14,24,34,28)(15,21,35,29)$ |
$ 14, 14, 7 $ | $360$ | $14$ | $( 1,29,15,33, 9,21,19, 2,26,13,35,10,22,20)( 3,28,14,32, 6,24,16, 4,30,11,34, 8,23,17)( 5,27,12,31, 7,25,18)$ |
$ 14, 14, 7 $ | $360$ | $14$ | $( 1,29,15,21,19,10,35, 2,26,13,22,20, 9,33)( 3,28,14,24,16, 8,34, 4,30,11,23, 17, 6,32)( 5,27,12,25,18, 7,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 2, 3)( 6, 9,10)(13,14,15)(16,19,20)(21,23,22)(26,29,30)(33,34,35)$ |
$ 6, 6, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $420$ | $6$ | $( 1, 2, 3)( 6,15,10,14, 9,13)( 7,12)( 8,11)(16,35,20,34,19,33)(17,32)(18,31) (21,23,22)(26,29,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 1, 2, 3)( 6,26,13)( 7,27,12)( 8,28,11)( 9,29,14)(10,30,15)(16,22,33) (17,24,32)(18,25,31)(19,21,34)(20,23,35)$ |
$ 12, 6, 4, 4, 3, 2, 2, 1, 1 $ | $840$ | $12$ | $( 1, 2, 3)( 6,19,10,16, 9,20)( 7,18)( 8,17)(11,24,32,28)(12,25,31,27) (13,23,35,29,14,22,33,30,15,21,34,26)$ |
$ 21, 7, 7 $ | $480$ | $21$ | $( 1,29,14,35,10,23,19, 2,30,15,33, 6,22,20, 3,26,13,34, 9,21,16) ( 4,28,11,32, 8,24,17)( 5,27,12,31, 7,25,18)$ |
$ 21, 7, 7 $ | $480$ | $21$ | $( 1,29,14,22,20, 6,35, 2,30,15,21,16, 9,33, 3,26,13,23,19,10,34) ( 4,28,11,24,17, 8,32)( 5,27,12,25,18, 7,31)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 8, 7, 9,10)(11,12,15,13,14)(16,17,18,19,20) (21,23,24,25,22)(26,29,30,28,27)(31,35,33,34,32)$ |
$ 10, 10, 5, 5, 5 $ | $252$ | $10$ | $( 1, 2, 3, 4, 5)( 6,11, 7,15,10,14, 8,12, 9,13)(16,32,18,35,20,34,17,31,19,33) (21,23,24,25,22)(26,29,30,28,27)$ |
$ 15, 15, 5 $ | $672$ | $15$ | $( 1, 2, 3, 4, 5)( 6,28,12, 9,29,14, 8,27,15,10,30,11, 7,26,13)(16,24,31,19,21, 34,17,25,35,20,23,32,18,22,33)$ |
$ 20, 10, 5 $ | $504$ | $20$ | $( 1, 2, 3, 4, 5)( 6,17, 7,19,10,16, 8,18, 9,20)(11,25,35,29,14,24,31,26,13,23, 32,27,15,21,34,28,12,22,33,30)$ |
$ 35 $ | $288$ | $35$ | $( 1,29,14,32, 7,22,20, 3,28,12,35,10,23,17, 5,26,13,34, 8,25,19, 2,30,11,31, 9,21,16, 4,27,15,33, 6,24,18)$ |
$ 35 $ | $288$ | $35$ | $( 1,29,14,24,18, 9,33, 3,28,12,22,20, 6,32, 5,26,13,23,17, 7,35, 2,30,11,25, 19,10,34, 4,27,15,21,16, 8,31)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8, 9,10)(11,15,13,14,12)(16,18,17,19,20) (21,23,25,24,22)(26,29,30,27,28)(31,32,35,33,34)$ |
$ 10, 10, 5, 5, 5 $ | $252$ | $10$ | $( 1, 2, 3, 5, 4)( 6,12, 8,15,10,14, 7,11, 9,13)(16,31,17,35,20,34,18,32,19,33) (21,23,25,24,22)(26,29,30,27,28)$ |
$ 15, 15, 5 $ | $672$ | $15$ | $( 1, 2, 3, 5, 4)( 6,27,11, 9,29,14, 7,28,15,10,30,12, 8,26,13)(16,25,32,19,21, 34,18,24,35,20,23,31,17,22,33)$ |
$ 20, 10, 5 $ | $504$ | $20$ | $( 1, 2, 3, 5, 4)( 6,18, 8,19,10,16, 7,17, 9,20)(11,22,33,30,12,24,35,29,14,25, 32,26,13,23,31,28,15,21,34,27)$ |
$ 35 $ | $288$ | $35$ | $( 1,29,14,31, 8,22,20, 3,27,11,35,10,23,18, 4,26,13,34, 7,24,19, 2,30,12,32, 9,21,16, 5,28,15,33, 6,25,17)$ |
$ 35 $ | $288$ | $35$ | $( 1,29,14,25,17, 9,33, 3,27,11,22,20, 6,31, 4,26,13,23,18, 8,35, 2,30,12,24, 19,10,34, 5,28,15,21,16, 7,32)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $10080=2^{5} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 10080.p | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);