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Magma
magma: G := TransitiveGroup(16, 44);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $44$ | 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: | (1,7,16,9,2,8,15,10)(3,13,6,12,4,14,5,11), (1,8)(2,7)(3,14)(4,13)(5,11)(6,12)(9,16)(10,15), (1,14,2,13)(3,7,4,8)(5,9,6,10)(11,16,12,15) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ 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
16T44, 16T47, 32T26Siblings 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 | Index | Representative |
1A | $1^{16}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{8}$ | $1$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
2B | $2^{8}$ | $2$ | $2$ | $8$ | $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$ |
2C | $2^{6},1^{4}$ | $4$ | $2$ | $6$ | $( 5, 6)( 7,10)( 8, 9)(11,13)(12,14)(15,16)$ |
2D | $2^{8}$ | $4$ | $2$ | $8$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$ |
4A1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5,15, 6,16)( 7,14, 8,13)( 9,11,10,12)$ |
4A-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5,16, 6,15)( 7,13, 8,14)( 9,12,10,11)$ |
4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1,16, 2,15)( 3, 6, 4, 5)( 7, 9, 8,10)(11,13,12,14)$ |
4C | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 6, 2, 5)( 3,15, 4,16)( 7,13, 8,14)( 9,11,10,12)$ |
4D | $4^{4}$ | $4$ | $4$ | $12$ | $( 1,13, 2,14)( 3, 8, 4, 7)( 5,10, 6, 9)(11,15,12,16)$ |
8A1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1,12,16,14, 2,11,15,13)( 3,10, 6, 7, 4, 9, 5, 8)$ |
8A-1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 9,15, 7, 2,10,16, 8)( 3,12, 5,13, 4,11, 6,14)$ |
8B1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1,11,16,13, 2,12,15,14)( 3, 9, 6, 8, 4,10, 5, 7)$ |
8B3 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 7,16, 9, 2, 8,15,10)( 3,13, 6,12, 4,14, 5,11)$ |
Malle's constant $a(G)$: $1/6$
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: | 32.42 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 4A1 | 4A-1 | 4B | 4C | 4D | 8A1 | 8A-1 | 8B1 | 8B3 | ||
Size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 4B | 4B | 4B | 4B | |
Type | |||||||||||||||
32.42.1a | R | ||||||||||||||
32.42.1b | R | ||||||||||||||
32.42.1c | R | ||||||||||||||
32.42.1d | R | ||||||||||||||
32.42.1e | R | ||||||||||||||
32.42.1f | R | ||||||||||||||
32.42.1g | R | ||||||||||||||
32.42.1h | R | ||||||||||||||
32.42.2a | R | ||||||||||||||
32.42.2b | R | ||||||||||||||
32.42.2c1 | C | ||||||||||||||
32.42.2c2 | C | ||||||||||||||
32.42.2c3 | C | ||||||||||||||
32.42.2c4 | C |
magma: CharacterTable(G);