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Properties

Label 16T1162
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 $t$ :  $1162$
Parity:  $1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (5,6)(7,8)(9,10)(13,14), (1,9,7,16,6,13,3,12)(2,10,8,15,5,14,4,11), (1,9,4,11,6,13,7,16)(2,10,3,12,5,14,8,15)
$|\Aut(F/K)|$:  $2$

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:  $C_8:C_2$ x 4, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $(C_8:C_2):C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_8:C_2)$ x 2, $C_2 \times (C_2^2:C_4)$
64:  $((C_8 : C_2):C_2):C_2$ x 4, 16T72, 16T76, 16T95
128:  16T227 x 2, 16T252
256:  16T485
512:  32T22852

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8:C_2$

Low degree siblings

16T1134 x 16, 16T1162 x 7, 16T1236 x 8, 32T35694 x 16, 32T35695 x 8, 32T35696 x 8, 32T35697 x 8, 32T35698 x 8, 32T35699 x 8, 32T35700 x 8, 32T35894 x 8, 32T35895 x 32, 32T35896 x 32, 32T35897 x 8, 32T35898 x 4, 32T35899 x 4, 32T35900 x 4, 32T35901 x 8, 32T35902 x 4, 32T35903 x 4, 32T35904 x 8, 32T35905 x 4, 32T35906 x 8, 32T36517 x 16, 32T36518 x 4, 32T36519 x 16, 32T36520 x 16, 32T36521 x 4, 32T36522 x 16, 32T36523 x 8, 32T36524 x 4, 32T36525 x 4, 32T36526 x 8, 32T36527 x 8, 32T36528 x 4, 32T36529 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 $ $8$ $2$ $( 5, 6)( 7, 8)( 9,10)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 5, 6)( 9,10)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(11,12)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7, 8)( 9,10)(13,14)$
$ 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, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 7, 6, 3)( 2, 8, 5, 4)( 9,16,13,12)(10,15,14,11)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 8, 6, 3)( 2, 7, 5, 4)( 9,16,14,11)(10,15,13,12)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 7, 5, 3)( 2, 8, 6, 4)( 9,15,14,11)(10,16,13,12)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3, 6, 7)( 2, 4, 5, 8)( 9,12,13,16)(10,11,14,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,12,14,15)(10,11,13,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,12,13,15)(10,11,14,16)$
$ 8, 8 $ $64$ $8$ $( 1, 9, 7,16, 6,13, 3,12)( 2,10, 8,15, 5,14, 4,11)$
$ 8, 8 $ $64$ $8$ $( 1,10, 7,16, 5,13, 3,12)( 2, 9, 8,15, 6,14, 4,11)$
$ 8, 8 $ $64$ $8$ $( 1,16, 3, 9, 6,12, 7,13)( 2,15, 4,10, 5,11, 8,14)$
$ 8, 8 $ $64$ $8$ $( 1,16, 3,10, 6,12, 8,13)( 2,15, 4, 9, 5,11, 7,14)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $4$ $( 9,14,10,13)(11,16,12,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 5, 6)( 7, 8)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 5, 6)( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3, 4)( 7, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 5, 6)( 9,14,10,13)(11,15,12,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 1, 2)( 5, 6)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 5, 6)( 9,14)(10,13)(11,16)(12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 7, 6, 3)( 2, 8, 5, 4)( 9,11,13,16)(10,12,14,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 8, 6, 3)( 2, 7, 5, 4)( 9,11,14,15)(10,12,13,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 8, 5, 3)( 2, 7, 6, 4)( 9,11,14,16)(10,12,13,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 7, 5, 3)( 2, 8, 6, 4)( 9,11,13,15)(10,12,14,16)$
$ 8, 8 $ $64$ $8$ $( 1, 9, 4,11, 6,13, 7,16)( 2,10, 3,12, 5,14, 8,15)$
$ 8, 8 $ $64$ $8$ $( 1,10, 3,12, 6,14, 7,16)( 2, 9, 4,11, 5,13, 8,15)$
$ 8, 8 $ $64$ $8$ $( 1,16, 7,13, 6,12, 4,10)( 2,15, 8,14, 5,11, 3, 9)$
$ 8, 8 $ $64$ $8$ $( 1,16, 8,13, 5,11, 3,10)( 2,15, 7,14, 6,12, 4, 9)$

Group invariants

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