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Properties

Label 16T1281
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 $t$ :  $1281$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $7$
Generators:  (1,6,10,13)(2,5,9,14)(3,8,11,16,4,7,12,15), (1,15,12,7)(2,16,11,8)(3,13,9,6)(4,14,10,5)
$|\Aut(F/K)|$:  $2$

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: $C_2^3: C_4$

Low degree siblings

16T1263 x 4, 16T1281 x 3, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
$ 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, 2, 2, 2, 2 $ $32$ $4$ $( 1,10)( 2, 9)( 3,11, 4,12)( 5,14)( 6,13)( 7,15, 8,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,10)( 2, 9)( 3,12, 4,11)( 5,14)( 6,13)( 7,16, 8,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 3, 2, 4)( 5, 8)( 6, 7)( 9,11,10,12)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,12)(10,11)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3)( 2, 4)( 5, 7, 6, 8)( 9,12)(10,11)(13,16,14,15)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)$
$ 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, 4, 4 $ $64$ $8$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,11,16, 4, 7,12,15)$
$ 8, 4, 4 $ $64$ $8$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 7,11,15, 4, 8,12,16)$
$ 8, 4, 4 $ $64$ $8$ $( 1,13,10, 6)( 2,14, 9, 5)( 3,16,12, 8, 4,15,11, 7)$
$ 8, 4, 4 $ $64$ $8$ $( 1,13,10, 6)( 2,14, 9, 5)( 3,15,12, 7, 4,16,11, 8)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $32$ $4$ $( 5, 8, 6, 7)(11,12)(13,15)(14,16)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)(11,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,16)( 6,15)( 7,14)( 8,13)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,15, 6,16)( 7,14, 8,13)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,15, 6,16)( 7,13, 8,14)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 5, 6)( 7, 8)( 9,11,10,12)(13,15)(14,16)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $4$ $( 3, 4)( 5, 6)( 9,12)(10,11)(13,16,14,15)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $4$ $( 1, 2)( 5, 6)( 9,12)(10,11)(13,15,14,16)$
$ 4, 2, 2, 2, 2, 2, 2 $ $16$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,10,12)(13,16)(14,15)$
$ 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 9,11,10,12)(13,16)(14,15)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 1, 2)( 3, 4)( 9,11,10,12)(13,15)(14,16)$
$ 8, 4, 4 $ $64$ $8$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,13, 8,15, 6,14, 7,16)$
$ 8, 4, 4 $ $64$ $8$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,16, 7,14, 6,15, 8,13)$
$ 8, 8 $ $64$ $8$ $( 1, 6, 9,16, 2, 5,10,15)( 3, 8,12,14, 4, 7,11,13)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1, 6, 9,15)( 2, 5,10,16)( 3, 7,12,14)( 4, 8,11,13)$
$ 8, 8 $ $64$ $8$ $( 1,13,11, 6, 2,14,12, 5)( 3,16, 9, 7, 4,15,10, 8)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,13,12, 5)( 2,14,11, 6)( 3,15,10, 7)( 4,16, 9, 8)$

Group invariants

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