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Properties

Label 16T1263
Order \(1024\)
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 $t$ :  $1263$
Parity:  $1$
Primitive:  No
Nilpotency class:  $7$
Generators:  (1,9,8,15,3,12,5,13)(2,10,7,16,4,11,6,14), (3,4)(5,6)(7,8)(13,16,14,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $QD_{16}$, $C_2^2:C_4$, $Q_{16}$
32:  $C_4\wr C_2$, $C_2^3 : C_4 $, 32T50
64:  $((C_8 : C_2):C_2):C_2$, 16T154, 16T161
128:  32T1744
256:  16T684
512:  32T28201

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $((C_8 : C_2):C_2):C_2$

Low degree siblings

16T1263 x 3, 16T1281 x 4, 32T36722 x 4, 32T36723 x 2, 32T36724 x 2, 32T36725 x 2, 32T36726 x 2, 32T36727 x 2, 32T36728 x 2, 32T36856 x 2, 32T36857 x 2, 32T36858 x 2, 32T36859 x 2, 32T36860 x 2, 32T36861 x 2, 32T55481 x 2, 32T56980 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 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,15,12,13)(10,16,11,14)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,13,12,15)(10,14,11,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3, 4)( 5, 6)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)(11,12)(13,14)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 1, 2)( 5, 6)( 9,12,10,11)(13,15,14,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$
$ 8, 8 $ $64$ $8$ $( 1, 9, 8,15, 3,12, 5,13)( 2,10, 7,16, 4,11, 6,14)$
$ 8, 8 $ $64$ $8$ $( 1,12, 8,13, 3, 9, 5,15)( 2,11, 7,14, 4,10, 6,16)$
$ 8, 8 $ $64$ $8$ $( 1,15, 5, 9, 3,13, 8,12)( 2,16, 6,10, 4,14, 7,11)$
$ 8, 8 $ $64$ $8$ $( 1,13, 5,12, 3,15, 8, 9)( 2,14, 6,11, 4,16, 7,10)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $32$ $4$ $( 3, 4)( 5, 8, 6, 7)(13,15)(14,16)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 4, 2, 3)( 7, 8)( 9,12)(10,11)(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,15)(10,16)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
$ 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)(13,15,14,16)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 3, 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(15,16)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 9,10)(11,12)(13,16,14,15)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 3, 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,14)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 5, 6)( 7, 8)(13,16,14,15)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $16$ $4$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15,14,16)$
$ 8, 4, 2, 2 $ $64$ $8$ $( 1, 8, 3, 6, 2, 7, 4, 5)( 9,15,10,16)(11,14)(12,13)$
$ 8, 4, 2, 2 $ $64$ $8$ $( 1, 8, 3, 6, 2, 7, 4, 5)( 9,15)(10,16)(11,14,12,13)$
$ 8, 8 $ $64$ $8$ $( 1, 9, 8,15, 2,10, 7,16)( 3,11, 6,14, 4,12, 5,13)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,12, 8,13)( 2,11, 7,14)( 3,10, 6,16)( 4, 9, 5,15)$
$ 8, 8 $ $64$ $8$ $( 1,15, 7,11, 2,16, 8,12)( 3,14, 6,10, 4,13, 5, 9)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,13, 8, 9)( 2,14, 7,10)( 3,16, 5,12)( 4,15, 6,11)$

Group invariants

Order:  $1024=2^{10}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.