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Magma
magma: G := TransitiveGroup(12, 272);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $272$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $M_{11}$ | ||
CHM label: | $M_{11}(12)$ | ||
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: | (1,7,3,10,5,9,6,12)(2,11,8,4), (1,6,3,9)(2,7,12,10,4,5,11,8) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Degree 4: None
Degree 6: None
Low degree siblings
11T6, 22T22Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $165$ | $2$ | $( 2,10)( 3, 5)( 6, 9)( 8,11)$ |
$ 4, 4, 2, 2 $ | $990$ | $4$ | $( 1, 7)( 2, 8,10,11)( 3, 6, 5, 9)( 4,12)$ |
$ 8, 4 $ | $990$ | $8$ | $( 1,12, 7, 4)( 2, 3, 8, 6,10, 5,11, 9)$ |
$ 8, 4 $ | $990$ | $8$ | $( 1, 4, 7,12)( 2, 9,11, 5,10, 6, 8, 3)$ |
$ 5, 5, 1, 1 $ | $1584$ | $5$ | $( 1, 3,11,12, 4)( 5, 6, 9, 8, 7)$ |
$ 11, 1 $ | $720$ | $11$ | $( 1, 2, 9, 3,12,11, 5,10, 7, 4, 8)$ |
$ 11, 1 $ | $720$ | $11$ | $( 1, 8, 4, 7,10, 5,11,12, 3, 9, 2)$ |
$ 3, 3, 3, 1, 1, 1 $ | $440$ | $3$ | $( 1,10, 6)( 2,12, 9)( 3, 4, 7)$ |
$ 6, 3, 2, 1 $ | $1320$ | $6$ | $( 1, 6,10)( 2, 7,12, 3, 9, 4)( 5, 8)$ |
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 3 3 3 . . . 1 1 3 2 1 . . . . . . 2 1 5 1 . . . . 1 . . . . 11 1 . . . . . 1 1 . . 1a 2a 4a 8a 8b 5a 11a 11b 3a 6a 2P 1a 1a 2a 4a 4a 5a 11b 11a 3a 3a 3P 1a 2a 4a 8a 8b 5a 11a 11b 1a 2a 5P 1a 2a 4a 8b 8a 1a 11a 11b 3a 6a 7P 1a 2a 4a 8b 8a 5a 11b 11a 3a 6a 11P 1a 2a 4a 8a 8b 5a 1a 1a 3a 6a X.1 1 1 1 1 1 1 1 1 1 1 X.2 10 2 2 . . . -1 -1 1 -1 X.3 10 -2 . A -A . -1 -1 1 1 X.4 10 -2 . -A A . -1 -1 1 1 X.5 11 3 -1 -1 -1 1 . . 2 . X.6 16 . . . . 1 B /B -2 . X.7 16 . . . . 1 /B B -2 . X.8 44 4 . . . -1 . . -1 1 X.9 45 -3 1 -1 -1 . 1 1 . . X.10 55 -1 -1 1 1 . . . 1 -1 A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 |
magma: CharacterTable(G);