Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1275$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (1,2)(3,4)(5,11,6,12)(7,9,8,10)(13,15)(14,16), (1,8,4,6)(2,7,3,5)(9,16,12,14)(10,15,11,13), (1,5,16,9)(2,6,15,10)(3,8,13,11)(4,7,14,12) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $D_{4}$ x 6, $C_2^3$ 16: $D_{8}$ x 2, $D_4\times C_2$ x 3 32: $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T29 64: $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T126 128: $C_2 \wr C_2\wr C_2$ x 2, 16T409 256: 16T689 512: 16T962 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_2 \wr C_2\wr C_2$
Low degree siblings
16T1253 x 8, 16T1265 x 8, 16T1275 x 15, 32T36646 x 4, 32T36647 x 8, 32T36648 x 16, 32T36649 x 4, 32T36650 x 8, 32T36651 x 8, 32T36652 x 8, 32T36653 x 8, 32T36733 x 8, 32T36734 x 8, 32T36735 x 8, 32T36736 x 8, 32T36737 x 8, 32T36738 x 8, 32T36739 x 8, 32T36740 x 4, 32T36741 x 8, 32T36742 x 8, 32T36743 x 4, 32T36744 x 4, 32T36745 x 16, 32T36746 x 4, 32T36747 x 4, 32T36748 x 8, 32T36749 x 4, 32T36750 x 8, 32T36820 x 8, 32T36821 x 8, 32T36822 x 8, 32T36823 x 8, 32T36824 x 8, 32T36825 x 8, 32T41957 x 8, 32T56530 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 2)( 3, 4)( 5,11, 6,12)( 7, 9, 8,10)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 5,11, 6,12)( 7, 9, 8,10)(13,16)(14,15)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,15, 3,13)( 2,16, 4,14)( 5,11, 8,10)( 6,12, 7, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 3)( 2, 4)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1,15)( 2,16)( 3,13)( 4,14)( 9,11)(10,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,15)( 2,16)( 3,13)( 4,14)( 5, 6)( 7, 8)( 9,12)(10,11)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,16,12,14)(10,15,11,13)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,16,11,13)(10,15,12,14)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,10,13, 7, 4,12,16, 5)( 2, 9,14, 8, 3,11,15, 6)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,12,13, 5, 4,10,16, 7)( 2,11,14, 6, 3, 9,15, 8)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,13,10,14)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $32$ | $2$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 3, 4)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 1, 2)( 5,11, 8, 9)( 6,12, 7,10)(15,16)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 3, 2, 4)( 5, 9, 7,12)( 6,10, 8,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,15, 2,16)( 3,14, 4,13)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5, 9)( 6,10)( 7,12)( 8,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,11)( 8,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 7, 8)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,11,10,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3)( 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3)( 2, 4)( 5, 9, 6,10)( 7,12, 8,11)(13,15)(14,16)$ |
| $ 4, 4, 4, 2, 1, 1 $ | $64$ | $4$ | $( 1,13, 3,15)( 2,14, 4,16)( 7, 8)( 9,11,10,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,15, 2,16)( 3,13, 4,14)( 5,11, 7,10)( 6,12, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 4)( 2, 3)( 5,11)( 6,12)( 7,10)( 8, 9)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 5, 9, 6,10)( 7,12, 8,11)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,11, 7,10)( 6,12, 8, 9)$ |
| $ 8, 4, 2, 2 $ | $64$ | $8$ | $( 1, 8, 3, 5, 2, 7, 4, 6)( 9,16,10,15)(11,14)(12,13)$ |
| $ 8, 4, 2, 2 $ | $64$ | $8$ | $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,13,10,14)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,10,13, 5)( 2, 9,14, 6)( 3,11,16, 7)( 4,12,15, 8)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,12,14, 8)( 2,11,13, 7)( 3, 9,15, 6)( 4,10,16, 5)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |