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Magma
magma: G := TransitiveGroup(16, 260);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $260$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_8:C_8$ | ||
| 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: | (1,16)(2,15)(3,9)(4,10)(5,12)(6,11)(7,14)(8,13), (1,10,7,16,6,13,4,11,2,9,8,15,5,14,3,12) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$ $16$: $D_{8}$, $C_8:C_2$, $QD_{16}$, $C_2^2:C_4$, $C_8\times C_2$ $32$: $C_4\wr C_2$, $C_2^2 : C_8$, 16T26 $64$: 32T272 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_4\wr C_2$
Low degree siblings
16T260, 32T600 x 2Siblings 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 | $2^{4},1^{8}$ | $2$ | $2$ | $4$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| 2C | $2^{8}$ | $8$ | $2$ | $8$ | $( 1,16)( 2,15)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,14)( 8,13)$ |
| 4A1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$ |
| 4A-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$ |
| 4B | $4^{4}$ | $2$ | $4$ | $12$ | $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$ |
| 4C1 | $4^{2},1^{8}$ | $2$ | $4$ | $6$ | $( 9,13,10,14)(11,16,12,15)$ |
| 4C-1 | $4^{2},1^{8}$ | $2$ | $4$ | $6$ | $(1,6,2,5)(3,7,4,8)$ |
| 4D1 | $4^{2},2^{4}$ | $2$ | $4$ | $10$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,15,12,16)$ |
| 4D-1 | $4^{2},2^{4}$ | $2$ | $4$ | $10$ | $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| 4E | $4^{4}$ | $8$ | $4$ | $12$ | $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,10, 8, 9)$ |
| 8A1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,13,15,10,11,14,16)$ |
| 8A-1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,16,14,11,10,15,13,12)$ |
| 8B1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,15,14,12,10,16,13,11)$ |
| 8B-1 | $8^{2}$ | $2$ | $8$ | $14$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$ |
| 8C | $8,2^{2},1^{4}$ | $4$ | $8$ | $9$ | $( 3, 4)( 7, 8)( 9,16,13,12,10,15,14,11)$ |
| 8D | $8^{2}$ | $4$ | $8$ | $14$ | $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,15,14,12,10,16,13,11)$ |
| 8E1 | $8,4^{2}$ | $4$ | $8$ | $13$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,11,14,15,10,12,13,16)$ |
| 8E-1 | $8,4^{2}$ | $4$ | $8$ | $13$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,12,14,16,10,11,13,15)$ |
| 8E3 | $8,2^{2},1^{4}$ | $4$ | $8$ | $9$ | $( 3, 4)( 7, 8)( 9,15,13,11,10,16,14,12)$ |
| 8E-3 | $8,2^{2},1^{4}$ | $4$ | $8$ | $9$ | $( 1, 3, 6, 7, 2, 4, 5, 8)(11,12)(15,16)$ |
| 8F1 | $8,4^{2}$ | $4$ | $8$ | $13$ | $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,14,10,13)(11,16,12,15)$ |
| 8F-1 | $8^{2}$ | $4$ | $8$ | $14$ | $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,11,13,16,10,12,14,15)$ |
| 8F3 | $8,4^{2}$ | $4$ | $8$ | $13$ | $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,13,10,14)(11,15,12,16)$ |
| 8F-3 | $8,2^{2},1^{4}$ | $4$ | $8$ | $9$ | $( 1, 3, 6, 7, 2, 4, 5, 8)( 9,10)(13,14)$ |
| 8G1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1,13, 6, 9, 2,14, 5,10)( 3,15, 7,12, 4,16, 8,11)$ |
| 8G-1 | $8^{2}$ | $8$ | $8$ | $14$ | $( 1, 9, 5,13, 2,10, 6,14)( 3,12, 8,15, 4,11, 7,16)$ |
| 16A1 | $16$ | $8$ | $16$ | $15$ | $( 1,15, 4,10, 5,11, 7,14, 2,16, 3, 9, 6,12, 8,13)$ |
| 16A-1 | $16$ | $8$ | $16$ | $15$ | $( 1, 9, 8,16, 6,14, 3,11, 2,10, 7,15, 5,13, 4,12)$ |
| 16A3 | $16$ | $8$ | $16$ | $15$ | $( 1,13, 7,11, 6, 9, 4,15, 2,14, 8,12, 5,10, 3,16)$ |
| 16A-3 | $16$ | $8$ | $16$ | $15$ | $( 1,11, 3,13, 5,16, 8,10, 2,12, 4,14, 6,15, 7, 9)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $128=2^{7}$ | 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: | $4$ | ||
| Label: | 128.68 | magma: IdentifyGroup(G);
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| Character table: | 32 x 32 character table |
magma: CharacterTable(G);