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Magma
magma: G := TransitiveGroup(35, 28);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_7$ | ||
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), (2,5,11)(3,7,15)(4,9,10)(6,13,14)(16,21,23)(17,18,22)(19,26,20)(25,30,32)(27,33,34)(28,31,35) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Degree 7: None
Low degree siblings
7T6, 15T47 x 2, 21T33, 42T294, 42T299Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $280$ | $3$ | $( 1, 4,33)( 3, 6,14)( 5,10,22)( 7,12,21)( 8,18,31)( 9,26,17)(11,28,34) (13,32,23)(15,25,29)(16,30,24)(20,27,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 1,22,28)( 4, 5,34)( 7,23,25)( 8, 9,27)(10,11,33)(12,13,29)(15,21,32) (16,24,30)(17,20,31)(18,26,35)$ |
$ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1,17,19, 3, 7,16, 2)( 4,11,12,20, 9,25,13)( 5,18,14, 8,23,30,28) ( 6,15,10,24,22,31,27)(21,33,29,34,35,32,26)$ |
$ 7, 7, 7, 7, 7 $ | $360$ | $7$ | $( 1, 2,16, 7, 3,19,17)( 4,13,25, 9,20,12,11)( 5,28,30,23, 8,14,18) ( 6,27,31,22,24,10,15)(21,26,32,35,34,29,33)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $504$ | $5$ | $( 1,24,23, 6,18)( 2,16, 7,26,22)( 3,19,17, 5,21)( 4,15, 8,20,34) ( 9,14,27,31,32)(10,13,33,28,30)(11,29,25,35,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,11)( 2,24)( 3,20)( 4,21)( 6,26)( 7,13)( 8,14)( 9,31)(10,29)(12,33)(17,27) (22,25)(23,28)(32,34)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $630$ | $4$ | $( 1,32,11,34)( 2,17,24,27)( 3,10,20,29)( 4,31,21, 9)( 6, 7,26,13)( 8,28,14,23) (12,25,33,22)(15,35)(16,19)(18,30)$ |
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $210$ | $6$ | $( 1, 3, 2,19,24, 8)( 4,18,21,33,31,25)( 5,27)( 6,16,26,29, 7,28)( 9,11) (10,35,23)(12,15,22,14,30,20)(17,32,34)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 2520.a | magma: IdentifyGroup(G);
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Character table: |
2 3 . . . 3 2 . 2 2 3 2 . . . 1 . 2 2 1 5 1 . . 1 . . . . . 7 1 1 1 . . . . . . 1a 7a 7b 5a 2a 4a 3a 3b 6a 2P 1a 7a 7b 5a 1a 2a 3a 3b 3b 3P 1a 7b 7a 5a 2a 4a 1a 1a 2a 5P 1a 7b 7a 1a 2a 4a 3a 3b 6a 7P 1a 1a 1a 5a 2a 4a 3a 3b 6a X.1 1 1 1 1 1 1 1 1 1 X.2 6 -1 -1 1 2 . . 3 -1 X.3 10 A /A . -2 . 1 1 1 X.4 10 /A A . -2 . 1 1 1 X.5 14 . . -1 2 . -1 2 2 X.6 14 . . -1 2 . 2 -1 -1 X.7 15 1 1 . -1 -1 . 3 -1 X.8 21 . . 1 1 -1 . -3 1 X.9 35 . . . -1 1 -1 -1 -1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |
magma: CharacterTable(G);