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Magma
magma: G := TransitiveGroup(16, 10);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2 : C_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $16$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,3,6,15)(2,4,5,16)(7,9,11,14)(8,10,12,13) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4
Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2
Low degree siblings
8T10 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 6,15)( 2, 4, 5,16)( 7, 9,11,14)( 8,10,12,13)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4,13, 7)( 2, 3,14, 8)( 5,15, 9,12)( 6,16,10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,11)( 8,12)( 9,14)(10,13)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7,13, 4)( 2, 8,14, 3)( 5,12, 9,15)( 6,11,10,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 8, 6,12)( 2, 7, 5,11)( 3,10,15,13)( 4, 9,16,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5, 9)( 6,10)(11,16)(12,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $16=2^{4}$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | $2$ | ||
Label: | 16.3 | magma: IdentifyGroup(G);
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Character table: |
2 4 3 3 3 3 4 3 3 4 4 1a 2a 4a 4b 2b 2c 4c 4d 2d 2e 2P 1a 1a 2c 2e 1a 1a 2e 2c 1a 1a 3P 1a 2a 4d 4c 2b 2c 4b 4a 2d 2e X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 1 1 X.3 1 -1 1 -1 -1 1 -1 1 1 1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 X.5 1 -1 A -A 1 -1 A -A 1 -1 X.6 1 -1 -A A 1 -1 -A A 1 -1 X.7 1 1 A A -1 -1 -A -A 1 -1 X.8 1 1 -A -A -1 -1 A A 1 -1 X.9 2 . . . . 2 . . -2 -2 X.10 2 . . . . -2 . . -2 2 A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);