# Properties

 Label 16T509 Order $$256$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2\times D_4^2.C_2$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $509$ Group : $C_2\times D_4^2.C_2$ Parity: $1$ Primitive: No Generators: (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,10)(8,9), (7,16)(8,15), (1,10)(2,9)(3,4)(5,6)(7,8)(11,12)(13,14)(15,16), (1,5)(2,6)(9,14)(10,13) $|\Aut(F/K)|$: $4$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $V_4$ x 35
8:  $D_{4}$ x 12, $C_2^3$ x 15
16:  $D_4\times C_2$ x 18, $C_2^4$
32:  $V_4 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T105
128:  $C_2 \wr C_2\wr C_2$ x 2, 16T245

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $V_4$, $D_{4}$ x 2

Degree 8: $D_4\times C_2$, $C_2 \wr C_2\wr C_2$ x 2

## Low degree siblings

16T509 x 31, 16T602 x 32, 32T2525 x 8, 32T2526 x 8, 32T2527 x 16, 32T2528 x 16, 32T2529 x 16, 32T2530 x 8, 32T2531 x 16, 32T2953 x 8, 32T2954 x 8, 32T2955 x 8, 32T4749 x 16, 32T6221 x 8, 32T6319 x 8

Siblings 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $2$ $( 7,16)( 8,15)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $2$ $( 5,14)( 6,13)( 7,16)( 8,15)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $2$ $( 3, 7)( 4, 8)(11,15)(12,16)$ $4, 4, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $4$ $( 3, 7,12,16)( 4, 8,11,15)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $8$ $2$ $( 3, 7)( 4, 8)( 5,14)( 6,13)(11,15)(12,16)$ $4, 4, 2, 2, 1, 1, 1, 1$ $8$ $4$ $( 3, 7,12,16)( 4, 8,11,15)( 5,14)( 6,13)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 3,12)( 4,11)( 7,16)( 8,15)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $4$ $2$ $( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,10)(11,12)(13,14)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10)(11,16)(12,15)(13,14)$ $4, 4, 2, 2, 2, 2$ $4$ $4$ $( 1, 2)( 3, 8,12,15)( 4, 7,11,16)( 5, 6)( 9,10)(13,14)$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5,13)( 6,14)( 9,10)(11,16)(12,15)$ $4, 4, 2, 2, 2, 2$ $8$ $4$ $( 1, 2)( 3, 8,12,15)( 4, 7,11,16)( 5,13)( 6,14)( 9,10)$ $2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 2)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$ $4, 4, 2, 2, 2, 2$ $16$ $4$ $( 1, 3)( 2, 4)( 5, 7,14,16)( 6, 8,13,15)( 9,12)(10,11)$ $4, 4, 4, 4$ $16$ $4$ $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,12,14,16)(10,11,13,15)$ $8, 8$ $16$ $8$ $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$ $4, 4, 4, 4$ $8$ $4$ $( 1, 3, 9,12)( 2, 4,10,11)( 5, 7,14,16)( 6, 8,13,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$ $4, 4, 2, 2, 2, 2$ $16$ $4$ $( 1, 4)( 2, 3)( 5, 8,14,15)( 6, 7,13,16)( 9,11)(10,12)$ $4, 4, 4, 4$ $16$ $4$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,11,14,15)(10,12,13,16)$ $8, 8$ $16$ $8$ $( 1, 4, 5, 8, 9,11,14,15)( 2, 3, 6, 7,10,12,13,16)$ $4, 4, 4, 4$ $8$ $4$ $( 1, 4, 9,11)( 2, 3,10,12)( 5, 8,14,15)( 6, 7,13,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$ $4, 4, 2, 2, 2, 2$ $8$ $4$ $( 1, 5)( 2, 6)( 3, 7,12,16)( 4, 8,11,15)( 9,14)(10,13)$ $4, 4, 4, 4$ $4$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 5)( 2, 6)( 3,12)( 4,11)( 7,16)( 8,15)( 9,14)(10,13)$ $4, 4, 2, 2, 2, 2$ $4$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3,12)( 4,11)( 7,16)( 8,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$ $4, 4, 2, 2, 2, 2$ $8$ $4$ $( 1, 6)( 2, 5)( 3, 8,12,15)( 4, 7,11,16)( 9,13)(10,14)$ $4, 4, 4, 4$ $4$ $4$ $( 1, 6, 9,13)( 2, 5,10,14)( 3, 8,12,15)( 4, 7,11,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 6)( 2, 5)( 3,11)( 4,12)( 7,15)( 8,16)( 9,13)(10,14)$ $4, 4, 2, 2, 2, 2$ $4$ $4$ $( 1, 6, 9,13)( 2, 5,10,14)( 3,11)( 4,12)( 7,15)( 8,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$

## Group invariants

 Order: $256=2^{8}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [256, 26531]
 Character table: Data not available.