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Properties

Label 16T261
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times C_2\wr C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $261$
Group :  $C_2\times C_2\wr C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,15,13,12)(2,16,14,11)(3,9,7,5)(4,10,8,6), (1,6,9,14)(2,5,10,13)(3,8,12,16)(4,7,11,15), (1,7,13,3,9,15,5,12)(2,8,14,4,10,16,6,11)
$|\Aut(F/K)|$:  $4$

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:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $((C_8 : C_2):C_2):C_2$ x 2

Low degree siblings

16T227 x 4, 16T259 x 8, 16T261 x 3, 16T273 x 4, 16T283 x 4, 32T506, 32T507 x 2, 32T508 x 4, 32T595 x 4, 32T596 x 8, 32T597 x 2, 32T598 x 4, 32T599 x 4, 32T601, 32T602 x 2, 32T633, 32T657, 32T1130 x 2, 32T1796

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 850]
Character table: Data not available.