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Properties

Label 16T1027
Order \(576\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1027$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,16,5,4,15,6)(7,14,9,8,13,10)(11,12), (1,9,3,14,16,11,2,10,4,13,15,12)(5,8,6,7)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$
36:  $C_3\times (C_3 : C_4)$
288:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $A_4\wr C_2$

Low degree siblings

12T159, 24T1487, 24T1488, 36T719, 36T724, 36T946, 36T947

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 $ $6$ $2$ $( 7,13)( 8,14)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 3, 5,15)( 4, 6,16)( 9,11,13)(10,12,14)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 2)( 3,16, 5, 4,15, 6)( 7,14, 9, 8,13,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,16)$
$ 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, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,10)( 8, 9)(11,14)(12,13)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 9,13,11)(10,14,12)$
$ 3, 3, 2, 2, 2, 2, 1, 1 $ $24$ $6$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 9,13,11)(10,14,12)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $16$ $3$ $( 3,15, 5)( 4,16, 6)( 9,11,13)(10,12,14)$
$ 6, 6, 2, 2 $ $16$ $6$ $( 1, 2)( 3,16, 5, 4,15, 6)( 7,14,11, 8,13,12)( 9,10)$
$ 6, 2, 2, 2, 2, 2 $ $8$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,11, 8, 9,12)(13,14)(15,16)$
$ 6, 2, 2, 2, 2, 2 $ $24$ $6$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,10,11, 8, 9,12)(13,14)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 9,11,13)(10,12,14)$
$ 3, 3, 2, 2, 2, 2, 1, 1 $ $24$ $6$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 9,11,13)(10,12,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $16$ $3$ $( 3, 5,15)( 4, 6,16)( 9,13,11)(10,14,12)$
$ 6, 2, 2, 2, 2, 2 $ $24$ $6$ $( 1, 2)( 3,16, 5, 4,15, 6)( 7,14)( 8,13)( 9,12)(10,11)$
$ 6, 2, 2, 2, 2, 2 $ $8$ $6$ $( 1, 2)( 3,16, 5, 4,15, 6)( 7, 8)( 9,10)(11,12)(13,14)$
$ 6, 6, 2, 2 $ $16$ $6$ $( 1, 2)( 3, 6,15, 4, 5,16)( 7,12,13, 8,11,14)( 9,10)$
$ 12, 4 $ $48$ $12$ $( 1, 9, 3,14,16,11, 2,10, 4,13,15,12)( 5, 8, 6, 7)$
$ 12, 4 $ $48$ $12$ $( 1, 8, 3, 9,16,12, 2, 7, 4,10,15,11)( 5,13, 6,14)$
$ 12, 4 $ $48$ $12$ $( 1,13,15,10, 4,11, 2,14,16, 9, 3,12)( 5, 8, 6, 7)$
$ 12, 4 $ $48$ $12$ $( 1,10,15, 7, 4,12, 2, 9,16, 8, 3,11)( 5,13, 6,14)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5, 8, 6, 7)(13,15,14,16)$
$ 4, 4, 4, 4 $ $36$ $4$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,14,16, 7)( 6,13,15, 8)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1,12, 2,11)( 3, 7, 4, 8)( 5,13, 6,14)( 9,16,10,15)$
$ 4, 4, 4, 4 $ $36$ $4$ $( 1,10,15,11)( 2, 9,16,12)( 3,13, 6, 8)( 4,14, 5, 7)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8278]
Character table: Data not available.