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Properties

Label 16T193
Order \(96\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\GL(2,Z/4)$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $193$
Group :  $\GL(2,Z/4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,8,15)(2,13,7,16)(3,9,5,12)(4,10,6,11), (1,8,4)(2,7,3)(9,14,12,10,13,11)(15,16)
$|\Aut(F/K)|$:  $2$

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: $S_4$

Low degree siblings

12T49 x 2, 12T50, 12T52, 16T186, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $8$ $3$ $( 3, 5, 7)( 4, 6, 8)( 9,13,16)(10,14,15)$
$ 6, 3, 3, 2, 1, 1 $ $8$ $6$ $( 3, 5, 7)( 4, 6, 8)( 9,14,16,10,13,15)(11,12)$
$ 6, 3, 3, 2, 1, 1 $ $8$ $6$ $( 3, 7, 5)( 4, 8, 6)( 9,15,13,10,16,14)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 6, 6, 2, 2 $ $8$ $6$ $( 1, 2)( 3, 6, 7, 4, 5, 8)( 9,14,16,10,13,15)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,15)(10,16)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,16)(10,15)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,16)(10,15)(11,14)(12,13)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 9, 5,14)( 2,10, 6,13)( 3,11, 8,16)( 4,12, 7,15)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 9, 6,13)( 2,10, 5,14)( 3,11, 7,15)( 4,12, 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 9)( 2,10)( 3,14)( 4,13)( 5,15)( 6,16)( 7,11)( 8,12)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 9, 2,10)( 3,14, 4,13)( 5,15, 6,16)( 7,11, 8,12)$

Group invariants

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

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

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

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