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Magma
magma: G := TransitiveGroup(16, 50);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Q_{16}:C_2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8,2,7)(3,6,4,5)(9,12,10,11)(13,15,14,16), (9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,11)(4,12)(5,15)(6,16)(7,10)(8,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ 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
16T32, 32T18Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$1^{16}$ | $1$ | $1$ | $()$ | |
$2^{4},1^{8}$ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ | |
$2^{8}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$ | |
$4^{4}$ | $4$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ | |
$4^{4}$ | $4$ | $4$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ | |
$8^{2}$ | $4$ | $8$ | $( 1, 9, 3,16, 2,10, 4,15)( 5,11, 7,13, 6,12, 8,14)$ | |
$8^{2}$ | $4$ | $8$ | $( 1, 9, 4,15, 2,10, 3,16)( 5,11, 8,14, 6,12, 7,13)$ | |
$4^{4}$ | $4$ | $4$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,10, 6, 9)( 7,16, 8,15)$ | |
$2^{8}$ | $4$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,16)( 8,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | 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: | 32.44 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | ||
Size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||||||
32.44.1a | R | |||||||||||
32.44.1b | R | |||||||||||
32.44.1c | R | |||||||||||
32.44.1d | R | |||||||||||
32.44.1e | R | |||||||||||
32.44.1f | R | |||||||||||
32.44.1g | R | |||||||||||
32.44.1h | R | |||||||||||
32.44.2a | R | |||||||||||
32.44.2b | R | |||||||||||
32.44.4a | S |
magma: CharacterTable(G);