Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1518$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4,2,3)(5,12,16,8,10,13,6,11,15,7,9,14), (1,7,12)(2,8,11)(3,5,9,4,6,10)(13,15)(14,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 12: $A_4$, $C_6\times C_2$ 24: $A_4\times C_2$ x 3 48: $C_2^2 \times A_4$ 96: $C_2^4:C_6$ 192: $C_2\wr A_4$ x 2, 12T87 384: 16T722 768: 24T2253 1536: 24T3674 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $A_4$
Degree 8: $C_2\wr A_4$
Low degree siblings
16T1518 x 3, 32T205642 x 2, 32T205643 x 4Siblings 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, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$ |
| $ 8, 8 $ | $96$ | $8$ | $( 1,12, 4,10, 2,11, 3, 9)( 5,13, 7,16, 6,14, 8,15)$ |
| $ 4, 4, 4, 4 $ | $48$ | $4$ | $( 1,14, 2,13)( 3,15, 4,16)( 5,10, 6, 9)( 7,12, 8,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $48$ | $2$ | $( 1,13)( 2,14)( 3,16)( 4,15)( 5, 9)( 6,10)( 7,11)( 8,12)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 5, 8, 6, 7)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 1, 4, 2, 3)( 9,12,10,11)(13,14)(15,16)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 5,10,15)( 6, 9,16)( 7,12,13)( 8,11,14)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,15, 6,10,16)( 7,11,13, 8,12,14)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 4, 2, 3)( 5,11,15, 7,10,13, 6,12,16, 8, 9,14)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 5,15,10)( 6,16, 9)( 7,13,12)( 8,14,11)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 4)( 5,16,10, 6,15, 9)( 7,14,12, 8,13,11)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 4, 2, 3)( 5,13,10, 7,16,12, 6,14, 9, 8,15,11)$ |
| $ 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 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1,12, 4,10, 2,11, 3, 9)( 5,13, 8,15, 6,14, 7,16)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1,10, 3,12, 2, 9, 4,11)( 5,16, 7,14, 6,15, 8,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $96$ | $4$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,10, 6, 9)( 7,12, 8,11)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 5, 8, 6, 7)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $64$ | $6$ | $( 5,10,15, 6, 9,16)( 7,12,13, 8,11,14)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,15)( 6,10,16)( 7,11,13)( 8,12,14)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 4, 2, 3)( 5,11,15, 8, 9,14, 6,12,16, 7,10,13)$ |
| $ 6, 6, 1, 1, 1, 1 $ | $64$ | $6$ | $( 5,15, 9, 6,16,10)( 7,13,11, 8,14,12)$ |
| $ 3, 3, 3, 3, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 4)( 5,16, 9)( 6,15,10)( 7,14,11)( 8,13,12)$ |
| $ 12, 4 $ | $128$ | $12$ | $( 1, 4, 2, 3)( 5,13, 9, 8,15,12, 6,14,10, 7,16,11)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(15,16)$ |
| $ 8, 2, 2, 2, 2 $ | $96$ | $8$ | $( 1,12, 3,10, 2,11, 4, 9)( 5,13)( 6,14)( 7,15)( 8,16)$ |
| $ 8, 4, 4 $ | $96$ | $8$ | $( 1,11, 4,10, 2,12, 3, 9)( 5,13, 6,14)( 7,15, 8,16)$ |
| $ 6, 6, 2, 1, 1 $ | $128$ | $6$ | $( 3, 4)( 5,10,15, 7,11,13)( 6, 9,16, 8,12,14)$ |
| $ 6, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 4)( 2, 3)( 5,11,16)( 6,12,15)( 7, 9,14, 8,10,13)$ |
| $ 6, 6, 2, 1, 1 $ | $128$ | $6$ | $( 3, 4)( 5,15,12, 8,13,10)( 6,16,11, 7,14, 9)$ |
| $ 6, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 4)( 2, 3)( 5,13,11, 6,14,12)( 7,15,10)( 8,16, 9)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,14)$ |
| $ 8, 4, 4 $ | $96$ | $8$ | $( 1,12, 3,10, 2,11, 4, 9)( 5,13, 6,14)( 7,15, 8,16)$ |
| $ 8, 2, 2, 2, 2 $ | $96$ | $8$ | $( 1,11, 4,10, 2,12, 3, 9)( 5,13)( 6,14)( 7,15)( 8,16)$ |
| $ 6, 6, 2, 1, 1 $ | $128$ | $6$ | $( 3, 4)( 5,10,15, 8,12,14)( 6, 9,16, 7,11,13)$ |
| $ 6, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 4)( 2, 3)( 5,11,16, 6,12,15)( 7, 9,14)( 8,10,13)$ |
| $ 6, 6, 2, 1, 1 $ | $128$ | $6$ | $( 3, 4)( 5,15,11, 7,14,10)( 6,16,12, 8,13, 9)$ |
| $ 6, 3, 3, 2, 2 $ | $128$ | $6$ | $( 1, 4)( 2, 3)( 5,13,12)( 6,14,11)( 7,15, 9, 8,16,10)$ |
Group invariants
| Order: | $3072=2^{10} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |