Properties

Label 16T211
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4.C_2^2:D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $211$
Group :  $C_4.C_2^2:D_4$
Parity:  $1$
Primitive:  No
Generators:  (1,6)(2,5)(3,8)(4,7)(9,13)(10,14)(11,16)(12,15), (1,9,2,10)(3,11,4,12)(5,13,6,14)(7,16,8,15), (1,4,6,7)(2,3,5,8)(9,14)(10,13)(11,15)(12,16)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $V_4$ x 7
8:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
32:  $C_4\wr C_2$ x 4, $V_4 \wr C_2 $, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
64:  16T111 x 2, 32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_4\wr C_2$ x 2, $V_4 \wr C_2 $

Low degree siblings

16T211 x 15, 32T461 x 4, 32T462 x 8, 32T1895 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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,11,14,15)(10,12,13,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,12,14,16)(10,11,13,15)$
$ 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 $ $2$ $2$ $( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,16,14,12)(10,15,13,11)$
$ 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 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,15)(10,12,13,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,16)(10,11,13,15)$
$ 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,14,12)(10,15,13,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,11,14,15)(10,12,13,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,12,14,16)(10,11,13,15)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,16,14,12)(10,15,13,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,12,14,16)(10,11,13,15)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,16,14,12)(10,15,13,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16,14,12)(10,15,13,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,16,14,12)(10,15,13,11)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 6, 4)( 2, 8, 5, 3)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 7, 6, 4)( 2, 8, 5, 3)( 9,16,14,12)(10,15,13,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 8, 6, 3)( 2, 7, 5, 4)( 9,15,14,11)(10,16,13,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,16, 8,15)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 3,11, 6,14, 8,15)( 2,10, 4,12, 5,13, 7,16)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 4,12, 6,14, 7,16)( 2,10, 3,11, 5,13, 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 5,13)( 2,10, 6,14)( 3,11, 7,16)( 4,12, 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 6,14)( 2,10, 5,13)( 3,11, 8,15)( 4,12, 7,16)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 7,16, 6,14, 4,12)( 2,10, 8,15, 5,13, 3,11)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 8,15, 6,14, 3,11)( 2,10, 7,16, 5,13, 4,12)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 628]
Character table: Data not available.