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Properties

Label 16T322
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times D_4).C_2^3$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $322$
Group :  $(C_2\times D_4).C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,4,2,3)(9,12,10,11), (1,5,11,14)(2,6,12,13)(3,7,9,15)(4,8,10,16), (1,14,11,6)(2,13,12,5)(3,15,9,8)(4,16,10,7)
$|\Aut(F/K)|$:  $4$

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 4, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
64:  16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$

Low degree siblings

16T293 x 4, 16T322, 32T678 x 2, 32T679 x 2, 32T680 x 2, 32T738, 32T739, 32T1138 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 $ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 5, 7, 6, 8)(13,15,14,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $4$ $4$ $( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $4$ $4$ $( 5, 8, 6, 7)( 9,10)(11,12)(13,15,14,16)$
$ 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, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$
$ 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 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 9,15)( 2, 6,10,16)( 3, 7,12,13)( 4, 8,11,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 9,16)( 2, 6,10,15)( 3, 7,12,14)( 4, 8,11,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5,11,14)( 2, 6,12,13)( 3, 7, 9,15)( 4, 8,10,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5,11,13)( 2, 6,12,14)( 3, 7, 9,16)( 4, 8,10,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,13)( 8,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,13, 6,14)( 7,16, 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13,11, 5)( 2,14,12, 6)( 3,16, 9, 7)( 4,15,10, 8)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13,12, 5)( 2,14,11, 6)( 3,16,10, 7)( 4,15, 9, 8)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13,10, 8)( 2,14, 9, 7)( 3,16,11, 5)( 4,15,12, 6)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 9, 8)( 2,14,10, 7)( 3,16,12, 5)( 4,15,11, 6)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 858]
Character table: Data not available.