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Properties

Label 16T875
Order \(512\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $875$
Parity:  $1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,13,5,11)(2,14,6,12)(3,16,7,10)(4,15,8,9), (1,6,3,8,2,5,4,7)(9,12)(10,11)(13,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 6, $C_2^2$
8:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
16:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
32:  $C_2^3 : C_4 $ x 6, 32T41
64:  16T77 x 3
128:  16T323
256:  32T7258

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2^3 : C_4 $

Low degree siblings

16T875 x 3, 32T10318 x 2, 32T10319 x 2, 32T10320 x 2, 32T21826 x 2, 32T21827 x 2, 32T26493, 32T33257 x 2

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

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 1645]
Character table: Data not available.