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Magma
magma: G := TransitiveGroup(16, 14);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Q_{16}$ | ||
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: | (1,13,2,14)(3,12,4,11)(5,9,6,10)(7,15,8,16), (1,12,2,11)(3,9,4,10)(5,8,6,7)(13,15,14,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$
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
Label | Cycle Type | Size | Order | Representative |
$1^{16}$ | $1$ | $1$ | $()$ | |
$2^{8}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$8^{2}$ | $2$ | $8$ | $( 1, 3, 5,15, 2, 4, 6,16)( 7,10,11,14, 8, 9,12,13)$ | |
$8^{2}$ | $2$ | $8$ | $( 1, 4, 5,16, 2, 3, 6,15)( 7, 9,11,13, 8,10,12,14)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 5, 2, 6)( 3,15, 4,16)( 7,11, 8,12)( 9,13,10,14)$ | |
$4^{4}$ | $4$ | $4$ | $( 1, 7, 2, 8)( 3,13, 4,14)( 5,12, 6,11)( 9,16,10,15)$ | |
$4^{4}$ | $4$ | $4$ | $( 1, 9, 2,10)( 3, 8, 4, 7)( 5,14, 6,13)(11,16,12,15)$ |
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: | $3$ | ||
Label: | 16.9 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 4A | 4B | 4C | 8A1 | 8A3 | ||
Size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | |
2 P | 1A | 1A | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||
16.9.1a | R | |||||||
16.9.1b | R | |||||||
16.9.1c | R | |||||||
16.9.1d | R | |||||||
16.9.2a | R | |||||||
16.9.2b1 | S | |||||||
16.9.2b2 | S |
magma: CharacterTable(G);