Show commands:
Magma
magma: G := TransitiveGroup(16, 45);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_8:C_2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15), (1,4,2,3)(5,16,6,15)(7,11,8,12)(9,14,10,13), (1,11)(2,12)(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 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: $D_4\times C_2$
Low degree siblings
8T15 x 2, 16T35, 16T38 x 2, 32T21Siblings 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 |
$1^{16}$ | $1$ | $1$ | $()$ | |
$2^{6},1^{4}$ | $4$ | $2$ | $( 5, 6)( 7,10)( 8, 9)(11,13)(12,14)(15,16)$ | |
$2^{8}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$4^{4}$ | $4$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,12, 8,11)( 9,13,10,14)$ | |
$4^{4}$ | $2$ | $4$ | $( 1, 3, 2, 4)( 5,16, 6,15)( 7,14, 8,13)( 9,11,10,12)$ | |
$2^{8}$ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$ | |
$8^{2}$ | $4$ | $8$ | $( 1, 7,16, 9, 2, 8,15,10)( 3,13, 6,12, 4,14, 5,11)$ | |
$2^{8}$ | $4$ | $2$ | $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$ | |
$8^{2}$ | $4$ | $8$ | $( 1,11,15,14, 2,12,16,13)( 3, 9, 5, 7, 4,10, 6, 8)$ | |
$2^{8}$ | $4$ | $2$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,15)(14,16)$ | |
$4^{4}$ | $2$ | $4$ | $( 1,15, 2,16)( 3, 5, 4, 6)( 7,10, 8, 9)(11,14,12,13)$ |
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: | 32.43 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | ||
Size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||||||
32.43.1a | R | |||||||||||
32.43.1b | R | |||||||||||
32.43.1c | R | |||||||||||
32.43.1d | R | |||||||||||
32.43.1e | R | |||||||||||
32.43.1f | R | |||||||||||
32.43.1g | R | |||||||||||
32.43.1h | R | |||||||||||
32.43.2a | R | |||||||||||
32.43.2b | R | |||||||||||
32.43.4a | R |
magma: CharacterTable(G);