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Properties

Label 16T61
Order \(48\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $61$
Group :  $C_2\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,12)(3,13)(4,14)(5,7)(6,8)(9,15)(10,16), (1,9,16,12,4,13)(2,10,15,11,3,14)(5,8)(6,7), (1,12)(2,11)(3,14,6,10,15,7)(4,13,5,9,16,8)
$|\Aut(F/K)|$:  $4$

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:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $S_4$

Degree 8: $S_4$, $S_4\times C_2$ x 2

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 24T46, 24T47, 24T48 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $8$ $3$ $( 3, 6,15)( 4, 5,16)( 7,14,10)( 8,13, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 3, 5,15)( 2, 4, 6,16)( 7,13,11, 9)( 8,14,12,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 7,16,10)( 2, 8,15, 9)( 3,12, 6,13)( 4,11, 5,14)$
$ 6, 6, 2, 2 $ $8$ $6$ $( 1, 8,16,12, 5,13)( 2, 7,15,11, 6,14)( 3,10)( 4, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 8)( 2, 7)( 3,14)( 4,13)( 5,12)( 6,11)( 9,16)(10,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,16)(14,15)$

Group invariants

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

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

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