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Properties

Label 16T937
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 $t$ :  $937$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,8,3,5)(2,7,4,6)(9,13)(10,14)(11,15,12,16), (1,8,2,7)(3,5)(4,6)(9,15,12,13)(10,16,11,14), (1,6,15,9,4,8,13,12,2,5,16,10,3,7,14,11)
$|\Aut(F/K)|$:  $2$

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
128:  16T227
256:  32T7382

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T937 x 7, 32T10662 x 4, 32T10663 x 4, 32T10664 x 4, 32T10665 x 4, 32T10666 x 4, 32T10667 x 4, 32T10668 x 4, 32T22175 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 $ $2$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)(11,12)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 1, 3, 2, 4)( 9,10)(11,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,13)(10,14)(11,15,12,16)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,13)(10,14)(11,16,12,15)$
$ 8, 8 $ $8$ $8$ $( 1,13, 4,16, 2,14, 3,15)( 5,10, 8,12, 6, 9, 7,11)$
$ 8, 8 $ $8$ $8$ $( 1,14, 4,15, 2,13, 3,16)( 5,10, 8,12, 6, 9, 7,11)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,13, 2,14)( 3,16, 4,15)( 5,10)( 6, 9)( 7,12)( 8,11)$
$ 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, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,10)(13,14)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 1, 3, 2, 4)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,14)(10,13)(11,16,12,15)$
$ 4, 4, 4, 2, 2 $ $32$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,14)(10,13)(11,15,12,16)$
$ 8, 8 $ $16$ $8$ $( 1,13, 3,15, 2,14, 4,16)( 5,10, 7,11, 6, 9, 8,12)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,10, 6, 9)( 7,12, 8,11)$
$ 8, 4, 1, 1, 1, 1 $ $16$ $8$ $( 5,11, 8, 9, 6,12, 7,10)(13,15,14,16)$
$ 8, 4, 2, 2 $ $16$ $8$ $( 1, 2)( 3, 4)( 5,11, 8, 9, 6,12, 7,10)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 3)( 2, 4)( 5,12, 6,11)( 7,10, 8, 9)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(15,16)$
$ 16 $ $32$ $16$ $( 1, 8,13,12, 4, 6,16, 9, 2, 7,14,11, 3, 5,15,10)$
$ 16 $ $32$ $16$ $( 1, 5,16, 9, 4, 7,14,12, 2, 6,15,10, 3, 8,13,11)$
$ 8, 4, 1, 1, 1, 1 $ $16$ $8$ $( 5,12, 7, 9, 6,11, 8,10)(13,16,14,15)$
$ 8, 4, 2, 2 $ $16$ $8$ $( 1, 2)( 3, 4)( 5,12, 7, 9, 6,11, 8,10)(13,15,14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 5,12, 6,11)( 7,10, 8, 9)(13,14)$
$ 16 $ $32$ $16$ $( 1, 8,14,12, 4, 6,15, 9, 2, 7,13,11, 3, 5,16,10)$
$ 16 $ $32$ $16$ $( 1, 5,15, 9, 4, 7,13,12, 2, 6,16,10, 3, 8,14,11)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 60692]
Character table: Data not available.