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Properties

Label 16T942
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 $t$ :  $942$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,5)(2,6)(3,7,4,8)(9,13,11,15)(10,14,12,16), (1,5,13,9,3,8,15,11,2,6,14,10,4,7,16,12), (1,3)(2,4)(5,8)(6,7)(11,12)(15,16)
$|\Aut(F/K)|$:  $2$

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:  16T567

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

16T942 x 3, 32T10688 x 2, 32T10689 x 2, 32T10690 x 2, 32T10691 x 2, 32T10692 x 2, 32T10693 x 2, 32T10694 x 2, 32T21132 x 2, 32T21136 x 2, 32T25895 x 2, 32T26102

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 60668]
Character table: Data not available.