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Properties

Label 16T406
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4^2:C_2^2.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $406$
Group :  $C_4^2:C_2^2.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,6,2,5)(3,8,4,7)(9,15,10,16)(11,13,12,14), (1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16), (1,10,3,12,2,9,4,11)(5,14,7,15,6,13,8,16)
$|\Aut(F/K)|$:  $4$

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 12, $C_2^3$
16:  $D_4\times C_2$ x 6, $Q_8:C_2$
32:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
64:  32T320

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

Low degree siblings

16T382 x 8, 16T406, 32T863 x 4, 32T864 x 4, 32T865 x 4, 32T901 x 2, 32T902, 32T903 x 2, 32T904

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 742]
Character table: Data not available.