Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $500$ | |
| Group : | $C_2^4.C_2^3.C_2$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $3$ | |
| Generators: | (1,13,4,11,5,10,8,15)(2,14,3,12,6,9,7,16), (1,9,5,14)(2,10,6,13)(3,12,7,16)(4,11,8,15), (1,6)(2,5)(3,8)(4,7)(9,12,14,16)(10,11,13,15) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_4$ x 4, $C_2^2$ x 7 8: $D_{4}$ x 12, $C_4\times C_2$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$ 32: $C_4\wr C_2$ x 2, $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3 64: $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T79, 16T106, 16T111, 16T138, 16T146 128: 32T1151, 32T1153, 32T1154 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_4\wr C_2$, $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$
Low degree siblings
16T500 x 7, 32T2494 x 4, 32T2495 x 2, 32T2496 x 2, 32T6982 x 2, 32T6986 x 2, 32T6996 x 2, 32T7040, 32T7041, 32T7042, 32T7044, 32T7162, 32T7163, 32T7165 x 2, 32T7314 x 2, 32T7315 x 2, 32T7362 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 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,12,14,16)(10,11,13,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 8)( 4, 7)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 8)( 4, 7)( 9,10)(11,15)(12,16)(13,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 3, 8)( 4, 7)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 3, 8)( 4, 7)( 9,12,10,11)(13,15,14,16)$ |
| $ 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 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,16)(10,11,13,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,15)(12,16)(13,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$ |
| $ 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 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,12,14,16)(10,11,13,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,15,14,11)(10,16,13,12)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,16,14,12)(10,15,13,11)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,16,14,12)(10,15,13,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,16, 8,15)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 3,12, 5,14, 7,16)( 2,10, 4,11, 6,13, 8,15)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 4,11, 5,14, 8,15)( 2,10, 3,12, 6,13, 7,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 5,14)( 2,10, 6,13)( 3,12, 7,16)( 4,11, 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 6,13)( 2,10, 5,14)( 3,12, 8,15)( 4,11, 7,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 3,15, 5,14, 7,11)( 2,10, 4,16, 6,13, 8,12)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 4,16, 5,14, 8,12)( 2,10, 3,15, 6,13, 7,11)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 5,14)( 2,10, 6,13)( 3,15, 7,11)( 4,16, 8,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 6,13)( 2,10, 5,14)( 3,15, 8,12)( 4,16, 7,11)$ |
Group invariants
| Order: | $256=2^{8}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [256, 1334] |
| Character table: Data not available. |