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Magma
magma: G := TransitiveGroup(35, 36);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_8$ | ||
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), (1,3,7)(2,6,5)(9,10,12)(11,14,13)(16,18,21)(17,22,23)(19,26,24)(25,31,28)(27,33,29)(30,35,32) | 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
8T49, 15T72 x 2, 28T433Siblings 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, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $2$ | $( 1,11)( 3,14)( 4,16)( 5,24)( 6,19)( 7,20)( 8,13)( 9,29)(10,28)(12,27)(17,25) (22,33)(23,31)(30,34)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $2520$ | $4$ | $( 1, 4,11,16)( 3, 6,14,19)( 5,10,24,28)( 7,12,20,27)( 8,25,13,17)( 9,23,29,31) (15,18)(21,35)(22,34,33,30)(26,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $112$ | $3$ | $( 1, 3,22)( 5, 6,34)( 7,23,17)( 8, 9,12)(11,14,33)(13,29,27)(15,21,18) (19,30,24)(20,31,25)(26,35,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1120$ | $3$ | $( 1,22, 3)( 4,10,16)( 5,33,19)( 6,11,30)( 7,18, 9)( 8,17,21)(12,23,15) (13,25,35)(14,24,34)(20,32,29)(26,27,31)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1, 4)( 3,16)( 6,24)( 7,21)( 8,18)( 9,17)(10,22)(11,34)(13,32)(14,30)(20,35) (25,29)$ |
$ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ | $3360$ | $6$ | $( 1,10, 3, 4,22,16)( 5,33,19)( 6,34,30,24,11,14)( 7, 8, 9,21,18,17)(12,23,15) (13,29,35,32,25,20)(26,27,31)$ |
$ 7, 7, 7, 7, 7 $ | $2880$ | $7$ | $( 1,32,23, 4,19,33,13)( 2,27,22, 5, 3,34, 9)( 6, 8,30,11,28,17,21) ( 7,15,35,10,16,18,26)(12,24,25,20,31,14,29)$ |
$ 7, 7, 7, 7, 7 $ | $2880$ | $7$ | $( 1,13,33,19, 4,23,32)( 2, 9,34, 3, 5,22,27)( 6,21,17,28,11,30, 8) ( 7,26,18,16,10,35,15)(12,29,14,31,20,25,24)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $1344$ | $5$ | $( 1,24,26,10,17)( 2,19,35,23,22)( 3,30, 6, 9,34)( 4,27,15,25,13) ( 5, 8,32, 7,12)(11,20,28,16,18)(14,31,29,21,33)$ |
$ 15, 15, 5 $ | $1344$ | $15$ | $( 1,22,20,24, 2,28,26,19,16,10,35,18,17,23,11)( 3,13, 8,30, 4,32, 6,27, 7, 9, 15,12,34,25, 5)(14,29,33,31,21)$ |
$ 15, 15, 5 $ | $1344$ | $15$ | $( 1,18,19,24,11,35,26,20,23,10,28,22,17,16, 2)( 3,12,27,30, 5,15, 6, 8,25, 9, 32,13,34, 7, 4)(14,29,33,31,21)$ |
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $1680$ | $6$ | $( 1, 5)( 3,34,28, 6,22, 4)( 7,17,31)( 8,27,29,13,12, 9)(10,19,33,16,14,30) (11,24)(15,32,35,26,18,21)(20,25,23)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ | $1260$ | $4$ | $( 2, 7,12,35)( 3, 8,29,28)( 4,23)( 5,17,18,11)( 6,10)( 9,34,16,25) (13,32,19,33)(14,26)(15,24,31,22)(20,21)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 20160.a | magma: IdentifyGroup(G);
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Character table: |
2 6 5 3 6 1 1 4 . . 2 . . . 2 3 2 1 . 1 2 1 . . . 2 1 1 1 1 5 1 . . . . . . . . 1 1 1 1 . 7 1 . . . . . . 1 1 . . . . . 1a 2a 4a 2b 3a 6a 4b 7a 7b 3b 5a 15a 15b 6b 2P 1a 1a 2a 1a 3a 3a 2b 7a 7b 3b 5a 15a 15b 3b 3P 1a 2a 4a 2b 1a 2b 4b 7b 7a 1a 5a 5a 5a 2a 5P 1a 2a 4a 2b 3a 6a 4b 7b 7a 3b 1a 3b 3b 6b 7P 1a 2a 4a 2b 3a 6a 4b 1a 1a 3b 5a 15b 15a 6b 11P 1a 2a 4a 2b 3a 6a 4b 7a 7b 3b 5a 15b 15a 6b 13P 1a 2a 4a 2b 3a 6a 4b 7b 7a 3b 5a 15b 15a 6b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 7 3 1 -1 1 -1 -1 . . 4 2 -1 -1 . X.3 14 2 . 6 2 . 2 . . -1 -1 -1 -1 -1 X.4 20 4 . 4 -1 1 . -1 -1 5 . . . 1 X.5 21 1 -1 -3 . . 1 . . 6 1 1 1 -2 X.6 21 1 -1 -3 . . 1 . . -3 1 B /B 1 X.7 21 1 -1 -3 . . 1 . . -3 1 /B B 1 X.8 28 4 . -4 1 -1 . . . 1 -2 1 1 1 X.9 35 -5 -1 3 2 . -1 . . 5 . . . 1 X.10 45 -3 1 -3 . . 1 A /A . . . . . X.11 45 -3 1 -3 . . 1 /A A . . . . . X.12 56 . . 8 -1 -1 . . . -4 1 1 1 . X.13 64 . . . -2 . . 1 1 4 -1 -1 -1 . X.14 70 2 . -2 1 1 -2 . . -5 . . . -1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 B = -E(15)-E(15)^2-E(15)^4-E(15)^8 = (-1-Sqrt(-15))/2 = -1-b15 |
magma: CharacterTable(G);