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Properties

Label 16T257
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^5.C_2.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $257$
Group :  $C_2^5.C_2.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,16,14,4,9,7,5,11)(2,15,13,3,10,8,6,12), (1,5,2,6)(3,8,12,15)(4,7,11,16)(9,14,10,13)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
32:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
64:  $((C_8 : C_2):C_2):C_2$ x 2, 16T84

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8:C_2$, $((C_8 : C_2):C_2):C_2$ x 2

Low degree siblings

16T228 x 8, 16T257 x 3, 16T258 x 4, 32T509 x 8, 32T510 x 4, 32T511 x 4, 32T512 x 2, 32T513 x 4, 32T591 x 2, 32T592 x 4, 32T593, 32T594, 32T1797

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 $ $4$ $2$ $( 5,13)( 6,14)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7,16)( 8,15)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5,13)( 6,14)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,11)( 4,12)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,14)( 6,13)( 7,16)( 8,15)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,16)( 8,15)( 9,10)(13,14)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5,15, 9,12,14, 8)( 2, 4, 6,16,10,11,13, 7)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 6, 8, 2, 4, 5, 7)( 9,12,13,15,10,11,14,16)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 6,16, 2, 4, 5,15)( 7,10,11,14, 8, 9,12,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 2, 6)( 3, 8,12,15)( 4, 7,11,16)( 9,14,10,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 8, 4, 7)(11,16,12,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 9,13)( 2, 5,10,14)( 3, 8,12,15)( 4, 7,11,16)$
$ 8, 8 $ $8$ $8$ $( 1, 7, 5, 4, 2, 8, 6, 3)( 9,16,14,11,10,15,13,12)$
$ 8, 8 $ $8$ $8$ $( 1, 7,13, 4, 2, 8,14, 3)( 5,12, 9,16, 6,11,10,15)$
$ 8, 8 $ $8$ $8$ $( 1, 7,14,11, 9,16, 5, 4)( 2, 8,13,12,10,15, 6, 3)$
$ 8, 8 $ $8$ $8$ $( 1, 7, 6,11, 9,16,13, 4)( 2, 8, 5,12,10,15,14, 3)$
$ 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:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 48]
Character table: Data not available.