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Properties

Label 16T1540
Order \(3072\)
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 $t$ :  $1540$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,2,11)(3,10,4,9), (1,8,16,9,3,6,13,11,2,7,15,10,4,5,14,12), (1,16,4,13,2,15,3,14)(5,8)(6,7)(9,12)(10,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$
192:  $V_4^2:(S_3\times C_2)$
384:  $C_2 \wr S_4$
768:  24T2409
1536:  24T4409

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 8: $C_2 \wr S_4$

Low degree siblings

16T1540 x 3, 32T205686 x 2, 32T205687 x 2, 32T205688 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 $ $6$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
$ 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, 4, 4 $ $6$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,15, 6,16)( 7,14, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $48$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,16)( 6,15)( 7,13)( 8,14)$
$ 8, 8 $ $96$ $8$ $( 1,15, 4,13, 2,16, 3,14)( 5,11, 7, 9, 6,12, 8,10)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 5, 7, 6, 8)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $128$ $3$ $( 5,12,16)( 6,11,15)( 7,10,13)( 8, 9,14)$
$ 6, 6, 2, 2 $ $128$ $6$ $( 1, 2)( 3, 4)( 5,11,15, 6,12,16)( 7, 9,14, 8,10,13)$
$ 12, 4 $ $256$ $12$ $( 1, 4, 2, 3)( 5,10,16, 7,11,13, 6, 9,15, 8,12,14)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $4$ $( 5,15, 6,16)( 7,14, 8,13)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $48$ $2$ $( 5,16)( 6,15)( 7,13)( 8,14)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1, 4, 2, 3)( 5,13, 6,14)( 7,15, 8,16)( 9,12,10,11)$
$ 4, 4, 2, 2, 2, 2 $ $48$ $4$ $( 1, 4, 2, 3)( 5,14)( 6,13)( 7,16)( 8,15)( 9,11,10,12)$
$ 8, 8 $ $192$ $8$ $( 1, 5,10,16, 2, 6, 9,15)( 3, 7,11,13, 4, 8,12,14)$
$ 8, 8 $ $192$ $8$ $( 1, 5, 9,16, 2, 6,10,15)( 3, 7,12,13, 4, 8,11,14)$
$ 8, 4, 1, 1, 1, 1 $ $96$ $8$ $( 5,14, 7,16, 6,13, 8,15)( 9,12,10,11)$
$ 8, 4, 2, 2 $ $96$ $8$ $( 1, 2)( 3, 4)( 5,14, 7,16, 6,13, 8,15)( 9,11,10,12)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $32$ $2$ $( 3, 4)( 7, 8)(11,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $32$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,14)$
$ 8, 2, 2, 2, 2 $ $96$ $8$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,15, 8,14, 6,16, 7,13)$
$ 8, 4, 4 $ $48$ $8$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,16, 7,14, 6,15, 8,13)$
$ 8, 4, 4 $ $48$ $8$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,16, 7,14, 6,15, 8,13)$
$ 6, 6, 2, 1, 1 $ $256$ $6$ $( 3, 4)( 5,12,15, 8,10,13)( 6,11,16, 7, 9,14)$
$ 6, 3, 3, 2, 2 $ $256$ $6$ $( 1, 4)( 2, 3)( 5,10,16, 6, 9,15)( 7,12,13)( 8,11,14)$
$ 8, 2, 2, 1, 1, 1, 1 $ $96$ $8$ $( 3, 4)( 5,15, 8,14, 6,16, 7,13)(11,12)$
$ 8, 2, 2, 2, 2 $ $96$ $8$ $( 1, 4)( 2, 3)( 5,13, 7,16, 6,14, 8,15)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $96$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 6)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 1, 1 $ $96$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5, 6)(13,16)(14,15)$
$ 16 $ $192$ $16$ $( 1, 5,10,16, 4, 7,12,13, 2, 6, 9,15, 3, 8,11,14)$
$ 16 $ $192$ $16$ $( 1, 5, 9,16, 4, 7,11,13, 2, 6,10,15, 3, 8,12,14)$

Group invariants

Order:  $3072=2^{10} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.