Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1046$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| 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) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|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
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $24$ | $4$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,15)( 6,16)( 7,14)( 8,13)$ |
| $ 4, 4, 4, 4 $ | $12$ | $4$ | $( 1,14, 2,13)( 3,16, 4,15)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,12)( 6,11)( 7, 9)( 8,10)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $32$ | $3$ | $( 5,12,13)( 6,11,14)( 7, 9,15)( 8,10,16)$ |
| $ 6, 6, 2, 2 $ | $32$ | $6$ | $( 1, 2)( 3, 4)( 5,11,14, 6,12,13)( 7,10,16, 8, 9,15)$ |
| $ 12, 4 $ | $32$ | $12$ | $( 1, 9, 5, 3,11, 7, 2,10, 6, 4,12, 8)(13,15,14,16)$ |
| $ 12, 4 $ | $32$ | $12$ | $( 1,10, 6, 4,11, 7, 2, 9, 5, 3,12, 8)(13,16,14,15)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 9,16,10,15)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $24$ | $2$ | $( 1, 2)( 3, 4)( 9,16)(10,15)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,10,15)(11,13,12,14)$ |
| $ 4, 4, 4, 4 $ | $48$ | $4$ | $( 1, 9, 6,16)( 2,10, 5,15)( 3,11, 8,13)( 4,12, 7,14)$ |
| $ 4, 4, 4, 4 $ | $12$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $24$ | $4$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $12$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,12,10,11)(13,16,14,15)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1,14, 6,12, 2,13, 5,11)( 3,16, 8, 9, 4,15, 7,10)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $12$ | $4$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 6,16)( 7,14, 8,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $24$ | $4$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $32$ | $6$ | $( 5,12,13, 6,11,14)( 7, 9,15, 8,10,16)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $32$ | $6$ | $( 1, 2)( 3, 4)( 5,11,14)( 6,12,13)( 7,10,16)( 8, 9,15)$ |
| $ 12, 4 $ | $32$ | $12$ | $( 1, 9, 5, 3,11, 7, 2,10, 6, 4,12, 8)(13,16,14,15)$ |
| $ 12, 4 $ | $32$ | $12$ | $( 1,10, 6, 4,11, 7, 2, 9, 5, 3,12, 8)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 9,16)(10,15)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 1, 2)( 3, 4)( 9,16,10,15)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,13)(12,14)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1, 9, 6,16, 2,10, 5,15)( 3,11, 8,13, 4,12, 7,14)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $48$ | $4$ | $( 1,14, 5,11)( 2,13, 6,12)( 3,16, 7,10)( 4,15, 8, 9)$ |
Group invariants
| Order: | $768=2^{8} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [768, 1088543] |
| Character table: Data not available. |