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Magma
magma: G := TransitiveGroup(12, 295);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $295$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $M_{12}$ | ||
CHM label: | $M(12)$ | ||
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: | (1,9,5,12,11,8,2,4)(6,10), (1,11,2,3,4)(5,8,12,6,7) | 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
12T295Siblings 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, 2, 2 $ | $396$ | $2$ | $( 1, 8)( 2, 3)( 4, 7)( 5,11)( 6,10)( 9,12)$ |
$ 3, 3, 3, 3 $ | $2640$ | $3$ | $( 1, 9, 2)( 3, 8,12)( 4, 6,11)( 5, 7,10)$ |
$ 6, 6 $ | $7920$ | $6$ | $( 1, 3, 9, 8, 2,12)( 4, 5, 6, 7,11,10)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $495$ | $2$ | $( 1, 2)( 3, 8)( 5,10)( 6,11)$ |
$ 4, 4, 2, 2 $ | $2970$ | $4$ | $( 1,11, 2, 6)( 3,10, 8, 5)( 4, 7)( 9,12)$ |
$ 8, 4 $ | $11880$ | $8$ | $( 1, 3,11,10, 2, 8, 6, 5)( 4,12, 7, 9)$ |
$ 4, 4, 1, 1, 1, 1 $ | $2970$ | $4$ | $( 1, 8, 2, 3)( 5, 6,10,11)$ |
$ 8, 2, 1, 1 $ | $11880$ | $8$ | $( 1, 5, 8, 6, 2,10, 3,11)( 4,12)$ |
$ 5, 5, 1, 1 $ | $9504$ | $5$ | $( 1,11, 5, 8, 6)( 3,10, 9, 7,12)$ |
$ 10, 2 $ | $9504$ | $10$ | $( 1, 9, 8, 3,11, 7, 6,10, 5,12)( 2, 4)$ |
$ 3, 3, 3, 1, 1, 1 $ | $1760$ | $3$ | $( 1, 6,12)( 2,10, 4)( 3, 9, 8)$ |
$ 6, 3, 2, 1 $ | $15840$ | $6$ | $( 1, 8, 6, 3,12, 9)( 2, 4,10)( 5,11)$ |
$ 11, 1 $ | $8640$ | $11$ | $( 1, 7, 9,10, 3, 8,11, 2,12, 5, 6)$ |
$ 11, 1 $ | $8640$ | $11$ | $( 1, 6, 5,12, 2,11, 8, 3,10, 9, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $95040=2^{6} \cdot 3^{3} \cdot 5 \cdot 11$ | 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: | 95040.a | magma: IdentifyGroup(G);
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Character table: |
2 6 6 5 3 4 2 2 5 1 1 3 1 1 . . 3 3 1 . . 1 2 1 . 3 1 . . . . . 5 1 . . . 1 . . . . . . 1 1 . . 11 1 . . . . . . . . . . . . 1 1 1a 2a 4a 8a 2b 3a 6a 4b 3b 6b 8b 5a 10a 11a 11b 2P 1a 1a 2a 4a 1a 3a 3a 2a 3b 3b 4b 5a 5a 11b 11a 3P 1a 2a 4a 8a 2b 1a 2b 4b 1a 2a 8b 5a 10a 11a 11b 5P 1a 2a 4a 8a 2b 3a 6a 4b 3b 6b 8b 1a 2b 11a 11b 7P 1a 2a 4a 8a 2b 3a 6a 4b 3b 6b 8b 5a 10a 11b 11a 11P 1a 2a 4a 8a 2b 3a 6a 4b 3b 6b 8b 5a 10a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 11 3 -1 -1 -1 -1 -1 3 2 . 1 1 -1 . . X.3 11 3 3 1 -1 -1 -1 -1 2 . -1 1 -1 . . X.4 16 . . . 4 1 1 . -2 . . 1 -1 A /A X.5 16 . . . 4 1 1 . -2 . . 1 -1 /A A X.6 45 -3 1 -1 5 3 -1 1 . . -1 . . 1 1 X.7 54 6 2 . 6 . . 2 . . . -1 1 -1 -1 X.8 55 7 -1 -1 -5 1 1 -1 1 1 -1 . . . . X.9 55 -1 -1 1 -5 1 1 3 1 -1 -1 . . . . X.10 55 -1 3 -1 -5 1 1 -1 1 -1 1 . . . . X.11 66 2 -2 . 6 . . -2 3 -1 . 1 1 . . X.12 99 3 -1 1 -1 3 -1 -1 . . 1 -1 -1 . . X.13 120 -8 . . . . . . 3 1 . . . -1 -1 X.14 144 . . . 4 -3 1 . . . . -1 -1 1 1 X.15 176 . . . -4 -1 -1 . -4 . . 1 1 . . A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 |
magma: CharacterTable(G);