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Properties

Label 16T1031
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 $t$ :  $1031$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,15,12)(2,13,16,11)(3,9,4,10)(5,7,6,8), (1,12,2,11)(3,7,16,9)(4,8,15,10)(5,13,6,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $A_4^2:C_4$

Low degree siblings

8T46, 12T160, 12T162, 16T1030, 18T182, 18T184, 24T1489, 24T1491, 24T1505, 24T1506 x 2, 24T1508, 32T34594, 36T764, 36T765, 36T766, 36T767, 36T964, 36T965

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 $ $6$ $2$ $( 1,16)( 2,15)( 3, 6)( 4, 5)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $64$ $3$ $( 3, 6,15)( 4, 5,16)( 9,12,13)(10,11,14)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $3$ $( 9,12,13)(10,11,14)$
$ 3, 3, 2, 2, 2, 2, 1, 1 $ $48$ $6$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 9,12,13)(10,11,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,15)( 2,16)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,13)(12,14)$
$ 4, 4, 2, 2, 2, 2 $ $72$ $4$ $( 1,15)( 2,16)( 3, 4)( 5, 6)( 7,13,10,12)( 8,14, 9,11)$
$ 4, 4, 4, 4 $ $36$ $4$ $( 1, 6,16, 3)( 2, 5,15, 4)( 7,13,10,12)( 8,14, 9,11)$
$ 4, 4, 4, 4 $ $72$ $4$ $( 1,14,15,12)( 2,13,16,11)( 3, 9, 4,10)( 5, 7, 6, 8)$
$ 8, 8 $ $72$ $8$ $( 1, 7, 6,13,16,10, 3,12)( 2, 8, 5,14,15, 9, 4,11)$
$ 4, 4, 4, 4 $ $72$ $4$ $( 1,12,15,14)( 2,11,16,13)( 3,10, 4, 9)( 5, 8, 6, 7)$
$ 8, 8 $ $72$ $8$ $( 1, 9, 3,11,16, 8, 6,14)( 2,10, 4,12,15, 7, 5,13)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8652]
Character table:    
      2  6  5  6  .  2  2  4  3  4  3  3  3  3
      3  2  1  .  2  2  1  .  .  .  .  .  .  .

        1a 2a 2b 3a 3b 6a 2c 4a 4b 4c 8a 4d 8b
     2P 1a 1a 1a 3a 3b 3b 1a 2a 2b 2c 4b 2c 4b
     3P 1a 2a 2b 1a 1a 2a 2c 4a 4b 4d 8b 4c 8a
     5P 1a 2a 2b 3a 3b 6a 2c 4a 4b 4c 8a 4d 8b
     7P 1a 2a 2b 3a 3b 6a 2c 4a 4b 4d 8b 4c 8a

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
X.3      1  1  1  1  1  1 -1 -1 -1  A  A -A -A
X.4      1  1  1  1  1  1 -1 -1 -1 -A -A  A  A
X.5      4  4  4 -2  1  1  .  .  .  .  .  .  .
X.6      4  4  4  1 -2 -2  .  .  .  .  .  .  .
X.7      6  2 -2  .  3 -1 -2  .  2  .  .  .  .
X.8      6  2 -2  .  3 -1  2  . -2  .  .  .  .
X.9      9 -3  1  .  .  .  1 -1  1 -1  1 -1  1
X.10     9 -3  1  .  .  .  1 -1  1  1 -1  1 -1
X.11     9 -3  1  .  .  . -1  1 -1  A -A -A  A
X.12     9 -3  1  .  .  . -1  1 -1 -A  A  A -A
X.13    12  4 -4  . -3  1  .  .  .  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i