Show commands:
Magma
magma: G := TransitiveGroup(16, 1046);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $1046$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $(C_2^3\times C_4):S_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,9,2,10)(3,11,4,12)(5,15)(6,16)(7,14)(8,13), (1,10)(2,9)(3,12)(4,11)(5,8,6,7)(13,15,14,16), (1,2)(3,4)(5,14,11)(6,13,12)(7,16,10)(8,15,9) | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 $24$: $S_4$, $S_3 \times C_2^2$ $48$: $S_4\times C_2$ x 3 $96$: 12T48 $192$: $V_4^2:(S_3\times C_2)$ $384$: 12T136 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $S_4$
Degree 8: $S_4\times C_2$
Low degree siblings
16T1046 x 3, 32T34667 x 2, 32T34668 x 2, 32T34669 x 2, 32T34798 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 | $2^{2},1^{12}$ | $4$ | $2$ | $2$ | $(1,2)(3,4)$ |
| 2C | $2^{6},1^{4}$ | $4$ | $2$ | $6$ | $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| 2D | $2^{4},1^{8}$ | $6$ | $2$ | $4$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| 2E | $2^{8}$ | $12$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,13)(12,14)$ |
| 2F | $2^{8}$ | $12$ | $2$ | $8$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,16)( 6,15)( 7,13)( 8,14)$ |
| 2G | $2^{4},1^{8}$ | $12$ | $2$ | $4$ | $( 9,15)(10,16)(11,14)(12,13)$ |
| 2H | $2^{8}$ | $12$ | $2$ | $8$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| 2I | $2^{6},1^{4}$ | $24$ | $2$ | $6$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12)$ |
| 3A | $3^{4},1^{4}$ | $32$ | $3$ | $8$ | $( 1,14,11)( 2,13,12)( 3,16,10)( 4,15, 9)$ |
| 4A1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$ |
| 4A-1 | $4^{4}$ | $1$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| 4B1 | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| 4B-1 | $4^{4}$ | $4$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$ |
| 4C | $4^{4}$ | $6$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$ |
| 4D | $4^{2},2^{4}$ | $12$ | $4$ | $10$ | $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,12,10,11)(13,16,14,15)$ |
| 4E | $4^{2},1^{8}$ | $12$ | $4$ | $6$ | $(1,5,2,6)(3,7,4,8)$ |
| 4F | $4^{4}$ | $12$ | $4$ | $12$ | $( 1,15, 2,16)( 3,14, 4,13)( 5, 9, 6,10)( 7,11, 8,12)$ |
| 4G | $4^{4}$ | $12$ | $4$ | $12$ | $( 1,11, 2,12)( 3,10, 4, 9)( 5,13, 6,14)( 7,15, 8,16)$ |
| 4H1 | $4^{4}$ | $12$ | $4$ | $12$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,14,10,13)(11,16,12,15)$ |
| 4H-1 | $4^{4}$ | $12$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,13,10,14)(11,15,12,16)$ |
| 4I1 | $4^{2},2^{4}$ | $12$ | $4$ | $10$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| 4I-1 | $4^{2},2^{4}$ | $12$ | $4$ | $10$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,11,10,12)(13,15,14,16)$ |
| 4J | $4^{2},2^{4}$ | $24$ | $4$ | $10$ | $( 1,14, 2,13)( 3,16, 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| 4K | $4^{2},2^{4}$ | $24$ | $4$ | $10$ | $( 1,16)( 2,15)( 3,13)( 4,14)( 5,10, 6, 9)( 7,12, 8,11)$ |
| 4L | $4^{2},2^{2},1^{4}$ | $24$ | $4$ | $8$ | $( 1, 2)( 3, 4)( 9,16,10,15)(11,13,12,14)$ |
| 4M | $4^{2},2^{4}$ | $24$ | $4$ | $10$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,14)(10,13)(11,16)(12,15)$ |
| 4N | $4^{4}$ | $24$ | $4$ | $12$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,12,10,11)(13,15,14,16)$ |
| 4O | $4^{4}$ | $48$ | $4$ | $12$ | $( 1,14, 5,11)( 2,13, 6,12)( 3,16, 7,10)( 4,15, 8, 9)$ |
| 4P | $4^{4}$ | $48$ | $4$ | $12$ | $( 1,15, 6,10)( 2,16, 5, 9)( 3,14, 8,12)( 4,13, 7,11)$ |
| 6A | $6^{2},2^{2}$ | $32$ | $6$ | $12$ | $( 1,13,11, 2,14,12)( 3,15,10, 4,16, 9)( 5, 6)( 7, 8)$ |
| 6B | $3^{4},2^{2}$ | $32$ | $6$ | $10$ | $( 1, 2)( 3, 4)( 5,11,14)( 6,12,13)( 7,10,16)( 8, 9,15)$ |
| 6C | $6^{2},1^{4}$ | $32$ | $6$ | $10$ | $( 5,12,14, 6,11,13)( 7, 9,16, 8,10,15)$ |
| 8A | $8^{2}$ | $48$ | $8$ | $14$ | $( 1,11, 5,14, 2,12, 6,13)( 3,10, 7,16, 4, 9, 8,15)$ |
| 8B | $8^{2}$ | $48$ | $8$ | $14$ | $( 1,10, 6,15, 2, 9, 5,16)( 3,12, 8,14, 4,11, 7,13)$ |
| 12A1 | $12,4$ | $32$ | $12$ | $14$ | $( 1, 3, 2, 4)( 5, 9,13, 7,11,15, 6,10,14, 8,12,16)$ |
| 12A-1 | $12,4$ | $32$ | $12$ | $14$ | $( 1, 4, 2, 3)( 5,10,13, 8,11,16, 6, 9,14, 7,12,15)$ |
| 12B1 | $12,4$ | $32$ | $12$ | $14$ | $( 1,16,12, 4,14,10, 2,15,11, 3,13, 9)( 5, 7, 6, 8)$ |
| 12B-1 | $12,4$ | $32$ | $12$ | $14$ | $( 1,15,12, 3,14, 9, 2,16,11, 4,13,10)( 5, 8, 6, 7)$ |
Malle's constant $a(G)$: $1/2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $768=2^{8} \cdot 3$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 768.1088543 | magma: IdentifyGroup(G);
| |
| Character table: | 40 x 40 character table |
magma: CharacterTable(G);