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Magma
magma: G := TransitiveGroup(16, 8);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4:C_4$ | ||
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: | (1,15,6,3)(2,16,5,4)(7,14,11,9)(8,13,12,10), (1,8,6,12)(2,7,5,11)(3,9,15,14)(4,10,16,13) | 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, $C_2^2$ $8$: $D_{4}$, $C_4\times C_2$, $Q_8$ 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 2
Degree 8: $C_4\times C_2$, $D_4$, $Q_8$
Low degree siblings
There are no siblings 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 6,15)( 2, 4, 5,16)( 7, 9,11,14)( 8,10,12,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,13)(10,14)$ |
$ 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 $ | $2$ | $4$ | $( 1, 7, 6,11)( 2, 8, 5,12)( 3,10,15,13)( 4, 9,16,14)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 6, 7)( 2,12, 5, 8)( 3,13,15,10)( 4,14,16, 9)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13, 2,14)( 3, 7, 4, 8)( 5, 9, 6,10)(11,16,12,15)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,15, 6, 3)( 2,16, 5, 4)( 7,14,11, 9)( 8,13,12,10)$ |
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: | 16.4 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 3 4 4 3 3 3 3 3 1a 2a 4a 2b 2c 4b 4c 4d 4e 4f 2P 1a 1a 2c 1a 1a 2c 2a 2c 2a 2c 3P 1a 2a 4f 2b 2c 4d 4c 4b 4e 4a 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 -1 -1 A -1 -A 1 -A X.6 1 1 -A -1 -1 -A -1 A 1 A X.7 1 1 A -1 -1 -A 1 A -1 -A X.8 1 1 -A -1 -1 A 1 -A -1 A X.9 2 -2 . 2 -2 . . . . . X.10 2 -2 . -2 2 . . . . . A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);