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Properties

Label 16T972
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 $t$ :  $972$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,13,2,14)(3,11,4,12)(5,10,6,9)(7,16,8,15), (1,2)(3,4)(5,6)(7,8), (1,8)(2,7)(3,5)(4,6)(9,13,10,14)(11,15,12,16), (1,13,5,10)(2,14,6,9)(3,11)(4,12)(7,16,8,15)
$|\Aut(F/K)|$:  $2$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$
128:  $C_2 \wr C_2\wr C_2$
256:  16T659

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T972 x 7, 32T10866 x 4, 32T10867 x 4, 32T10868 x 4, 32T10869 x 4, 32T10870 x 4, 32T10871 x 4, 32T10872 x 4, 32T21596 x 2, 32T21635 x 2, 32T21959 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
$ 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 $ $16$ $4$ $( 1,13, 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5, 9)( 6,10)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 7)( 4, 8)( 5, 6)(11,15)(12,16)(13,14)$
$ 8, 8 $ $4$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,14,15,10,11,13,16)$
$ 8, 8 $ $4$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $32$ $8$ $( 1,16, 5,12, 2,15, 6,11)( 3, 9, 8,14, 4,10, 7,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,16,12,15)$
$ 8, 2, 2, 2, 1, 1 $ $16$ $8$ $( 3, 7)( 4, 8)( 5, 6)( 9,15,13,12,10,16,14,11)$
$ 8, 2, 2, 2, 1, 1 $ $16$ $8$ $( 1, 2)( 3, 8)( 4, 7)( 9,16,13,11,10,15,14,12)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,13, 3,16)( 2,14, 4,15)( 5, 9, 7,11)( 6,10, 8,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,16)(10,15)(11,14)(12,13)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(11,15)(12,16)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14)(10,13)(15,16)$
$ 4, 4, 4, 2, 2 $ $64$ $4$ $( 1,13)( 2,14)( 3,11, 7,16)( 4,12, 8,15)( 5,10, 6, 9)$
$ 8, 2, 2, 2, 2 $ $16$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12)(10,11)(13,15)(14,16)$
$ 8, 2, 2, 2, 2 $ $16$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $32$ $2$ $( 3, 7)( 4, 8)( 5, 6)( 9,15)(10,16)(11,13)(12,14)$
$ 16 $ $32$ $16$ $( 1,13, 4,15, 5, 9, 7,11, 2,14, 3,16, 6,10, 8,12)$
$ 16 $ $32$ $16$ $( 1, 9, 3,12, 5,14, 8,15, 2,10, 4,11, 6,13, 7,16)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $8$ $( 9,11,14,16,10,12,13,15)$
$ 8, 2, 2, 2, 2 $ $4$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,15,10,11,13,16)$
$ 8, 4, 4 $ $8$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,16,13,11,10,15,14,12)$
$ 8, 4, 4 $ $8$ $8$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,15,13,12,10,16,14,11)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $8$ $( 9,12,14,15,10,11,13,16)$
$ 8, 2, 2, 2, 2 $ $4$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,16,10,12,13,15)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 60809]
Character table: Data not available.