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Properties

Label 16T707
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

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $707$
Group :  $C_2^4.C_2^3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,5)(2,6)(3,4)(9,13)(10,14)(11,12), (5,13)(6,14), (1,16,2,15)(3,14,4,13)(5,11,6,12)(7,9,8,10)
$|\Aut(F/K)|$:  $4$

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:  $QD_{16}$ x 2, $D_4\times C_2$ x 3
32:  $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T48
64:  $(((C_4 \times C_2): C_2):C_2):C_2$, 16T138, 16T155
128:  $C_2 \wr C_2\wr C_2$ x 2, 32T1557

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $QD_{16}$, $C_2 \wr C_2\wr C_2$ x 2

Low degree siblings

16T700 x 16, 16T707 x 15, 32T3354 x 8, 32T3355 x 16, 32T3356 x 4, 32T3357 x 8, 32T3358 x 8, 32T3378 x 4, 32T3379 x 8, 32T6122 x 4, 32T6223 x 4, 32T6225 x 8, 32T6315 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 7,15)( 8,16)$
$ 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, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 8)( 4, 7)( 5, 6)(11,16)(12,15)(13,14)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3, 8,11,16)( 4, 7,12,15)( 5, 6)(13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 8)( 4, 7)( 5,14)( 6,13)(11,16)(12,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3, 8,11,16)( 4, 7,12,15)( 5,14)( 6,13)$
$ 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 $ $4$ $2$ $( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,16)( 8,15)( 9,10)(11,12)(13,14)$
$ 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 $ $8$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3, 7,11,15)( 4, 8,12,16)( 5,13)( 6,14)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,16)( 8,15)( 9,10)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)( 9,10)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3, 2, 4)( 5, 7,14,16)( 6, 8,13,15)( 9,11,10,12)$
$ 8, 8 $ $16$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$
$ 8, 8 $ $16$ $8$ $( 1, 3, 5, 8,10,12,14,15)( 2, 4, 6, 7, 9,11,13,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3,10,12)( 2, 4, 9,11)( 5, 7, 6, 8)(13,15,14,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3,10,12)( 2, 4, 9,11)( 5, 7,14,16)( 6, 8,13,15)$
$ 8, 8 $ $16$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,12,13,15,10,11,14,16)$
$ 8, 8 $ $16$ $8$ $( 1, 4, 5, 7,10,11,14,16)( 2, 3, 6, 8, 9,12,13,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 8,12,15)( 4, 7,11,16)( 9,13,10,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 5)( 2, 6)( 3,12)( 4,11)( 7,15)( 8,16)( 9,13)(10,14)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5, 9,13)( 2, 6,10,14)( 3,12)( 4,11)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 5190]
Character table: Data not available.