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Properties

Label 16T62
Order \(48\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4:C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $62$
Group :  $A_4:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,4,2,16,3)(5,6)(7,12,10,8,11,9)(13,14), (1,8,3,13)(2,7,4,14)(5,10,16,12)(6,9,15,11)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $S_4$

Degree 8: $S_4$

Low degree siblings

12T27, 12T30, 24T51, 24T57

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 30]
Character table:    
      2  4  1  4  1  4  4  3  3  3  3
      3  1  1  1  1  .  .  .  .  .  .

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

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

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