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Magma
magma: G := TransitiveGroup(22, 22);
Group action invariants
Degree $n$: | $22$ | 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: | $M_{11}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11,3,19,14)(2,12,4,20,13)(5,17,16,9,7)(6,18,15,10,8), (1,22,15,4,20)(2,21,16,3,19)(5,12,14,9,8)(6,11,13,10,7) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 11: $M_{11}$
Low degree siblings
11T6, 12T272Siblings 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 $ | $1$ | $1$ | $()$ |
$ 11, 11 $ | $720$ | $11$ | $( 1, 9,16, 4,19,21,18,11,13, 6, 8)( 2,10,15, 3,20,22,17,12,14, 5, 7)$ |
$ 11, 11 $ | $720$ | $11$ | $( 1, 8, 6,13,11,18,21,19, 4,16, 9)( 2, 7, 5,14,12,17,22,20, 3,15,10)$ |
$ 5, 5, 5, 5, 1, 1 $ | $1584$ | $5$ | $( 1,22,19, 5,16)( 2,21,20, 6,15)( 3,11, 9,14, 7)( 4,12,10,13, 8)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $440$ | $3$ | $( 1,10,17)( 2, 9,18)( 3,21,14)( 4,22,13)( 7,15,11)( 8,16,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $165$ | $2$ | $( 1, 3)( 2, 4)( 5,19)( 6,20)( 9,22)(10,21)(13,18)(14,17)$ |
$ 4, 4, 4, 4, 2, 2, 1, 1 $ | $990$ | $4$ | $( 1,14, 3,17)( 2,13, 4,18)( 5,21,19,10)( 6,22,20, 9)( 7, 8)(11,12)$ |
$ 8, 8, 4, 2 $ | $990$ | $8$ | $( 1,10,14, 5, 3,21,17,19)( 2, 9,13, 6, 4,22,18,20)( 7,12, 8,11)(15,16)$ |
$ 8, 8, 4, 2 $ | $990$ | $8$ | $( 1,19,17,21, 3, 5,14,10)( 2,20,18,22, 4, 6,13, 9)( 7,11, 8,12)(15,16)$ |
$ 6, 6, 3, 3, 2, 2 $ | $1320$ | $6$ | $( 1,15,11,22, 4,17)( 2,16,12,21, 3,18)( 5,14)( 6,13)( 7,20,10)( 8,19, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$ | 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: | 7920.a | magma: IdentifyGroup(G);
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Character table: |
2 4 . . 4 1 1 . 3 3 3 3 2 . . 1 2 1 . . . . 5 1 . . . . . 1 . . . 11 1 1 1 . . . . . . . 1a 11a 11b 2a 3a 6a 5a 4a 8a 8b 2P 1a 11b 11a 1a 3a 3a 5a 2a 4a 4a 3P 1a 11a 11b 2a 1a 2a 5a 4a 8a 8b 5P 1a 11a 11b 2a 3a 6a 1a 4a 8b 8a 7P 1a 11b 11a 2a 3a 6a 5a 4a 8b 8a 11P 1a 1a 1a 2a 3a 6a 5a 4a 8a 8b X.1 1 1 1 1 1 1 1 1 1 1 X.2 10 -1 -1 2 1 -1 . 2 . . X.3 10 -1 -1 -2 1 1 . . B -B X.4 10 -1 -1 -2 1 1 . . -B B X.5 11 . . 3 2 . 1 -1 -1 -1 X.6 16 A /A . -2 . 1 . . . X.7 16 /A A . -2 . 1 . . . X.8 44 . . 4 -1 1 -1 . . . X.9 45 1 1 -3 . . . 1 -1 -1 X.10 55 . . -1 1 -1 . -1 1 1 A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 |
magma: CharacterTable(G);