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Properties

Label 16T186
Order \(96\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\GL(2,Z/4)$

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $186$
Group :  $\GL(2,Z/4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,2,14)(3,10,4,9)(5,7,6,8)(11,15,12,16), (1,5)(2,6)(3,4)(7,12)(8,11)(15,16), (1,3,16)(2,4,15)(9,14,12)(10,13,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $S_4$, $(C_6\times C_2):C_2$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$, $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

12T49 x 2, 12T50, 12T52, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 195]
Character table:    
      2  5  3  2  5  2  5  3  5  3  2  2  3  4  4
      3  1  .  1  1  1  .  .  .  .  1  1  .  .  1

        1a 2a 3a 2b 6a 2c 4a 2d 4b 6b 6c 4c 2e 2f
     2P 1a 1a 3a 1a 3a 1a 2c 1a 2b 3a 3a 2d 1a 1a
     3P 1a 2a 1a 2b 2b 2c 4a 2d 4b 2f 2f 4c 2e 2f
     5P 1a 2a 3a 2b 6a 2c 4a 2d 4b 6c 6b 4c 2e 2f

X.1      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  1
X.3      1 -1  1  1  1  1 -1  1  1 -1 -1  1 -1 -1
X.4      1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1
X.5      2  . -1  2 -1  2  .  2  . -1 -1  .  2  2
X.6      2  . -1  2 -1  2  .  2  .  1  1  . -2 -2
X.7      2  .  2 -2 -2  2  . -2  .  .  .  .  .  .
X.8      2  . -1 -2  1  2  . -2  .  A -A  .  .  .
X.9      2  . -1 -2  1  2  . -2  . -A  A  .  .  .
X.10     3 -1  .  3  . -1  1 -1 -1  .  .  1 -1  3
X.11     3 -1  .  3  . -1  1 -1  1  .  . -1  1 -3
X.12     3  1  .  3  . -1 -1 -1 -1  .  .  1  1 -3
X.13     3  1  .  3  . -1 -1 -1  1  .  . -1 -1  3
X.14     6  .  . -6  . -2  .  2  .  .  .  .  .  .

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3