↑

Properties

Label 16T516
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^3.C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $516$
Group :  $C_2^4.C_2^3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,11,14,5)(2,12,13,6)(3,9,16,7)(4,10,15,8), (1,16,2,15)(3,13,4,14)(5,6)(9,10), (1,15)(2,16)(3,14)(4,13)(5,12,6,11)(7,9,8,10)
$|\Aut(F/K)|$:  $2$

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:  $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3
64:  16T79, 16T101 x 2
128:  32T1350

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_2^2 \wr C_2$

Low degree siblings

16T516 x 7, 16T574 x 4, 32T2566 x 8, 32T2567 x 4, 32T2568 x 4, 32T2569 x 4, 32T2570 x 4, 32T2571 x 4, 32T2572 x 4, 32T2831 x 2, 32T2832 x 4, 32T2833 x 2, 32T2834 x 2, 32T7454 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 5715]
Character table: Data not available.