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Properties

Label 16T181
Order \(96\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $181$
Group :  $C_4\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,2,12)(3,9,4,10)(5,13,6,14)(7,16,8,15), (1,10,5,12,4,8,2,9,6,11,3,7)(13,16,14,15)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_4$, $S_3 \times C_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

12T53 x 2, 16T181, 24T129, 24T130, 24T167, 24T168 x 2, 24T169 x 2, 32T387

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 186]
Character table:    
      2  5  4  2  5  4  2  5  4  5  4  4   2  4  5  4   2  4  5  5  5
      3  1  .  1  1  .  1  .  .  .  .  .   1  .  .  .   1  .  .  1  1

        1a 2a 3a 2b 2c 6a 2d 4a 2e 4b 4c 12a 4d 4e 4f 12b 4g 4h 4i 4j
     2P 1a 1a 3a 1a 1a 3a 1a 2e 1a 2e 2b  6a 2d 2b 2b  6a 2d 2b 2b 2b
     3P 1a 2a 1a 2b 2c 2b 2d 4a 2e 4b 4f  4i 4g 4h 4c  4j 4d 4e 4j 4i
     5P 1a 2a 3a 2b 2c 6a 2d 4a 2e 4b 4c 12a 4d 4e 4f 12b 4g 4h 4i 4j
     7P 1a 2a 3a 2b 2c 6a 2d 4a 2e 4b 4f 12b 4g 4h 4c 12a 4d 4e 4j 4i
    11P 1a 2a 3a 2b 2c 6a 2d 4a 2e 4b 4f 12b 4g 4h 4c 12a 4d 4e 4j 4i

X.1      1  1  1  1  1  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 -1   1 -1  1  1  1
X.3      1 -1  1  1 -1  1  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 -1  -1 -1 -1 -1 -1
X.5      1 -1  1 -1  1 -1 -1  1  1 -1  A  -A  A -A -A   A -A  A  A -A
X.6      1 -1  1 -1  1 -1 -1  1  1 -1 -A   A -A  A  A  -A  A -A -A  A
X.7      1  1  1 -1 -1 -1 -1 -1  1  1  A   A  A  A -A  -A -A -A -A  A
X.8      1  1  1 -1 -1 -1 -1 -1  1  1 -A  -A -A -A  A   A  A  A  A -A
X.9      2  . -1  2  . -1  2  .  2  .  .  -1  .  2  .  -1  .  2  2  2
X.10     2  . -1  2  . -1  2  .  2  .  .   1  . -2  .   1  . -2 -2 -2
X.11     2  . -1 -2  .  1 -2  .  2  .  .   A  .  B  .  -A  . -B -B  B
X.12     2  . -1 -2  .  1 -2  .  2  .  .  -A  . -B  .   A  .  B  B -B
X.13     3 -1  .  3 -1  . -1  1 -1  1 -1   .  1 -1 -1   .  1 -1  3  3
X.14     3 -1  .  3 -1  . -1  1 -1  1  1   . -1  1  1   . -1  1 -3 -3
X.15     3  1  .  3  1  . -1 -1 -1 -1 -1   .  1  1 -1   .  1  1 -3 -3
X.16     3  1  .  3  1  . -1 -1 -1 -1  1   . -1 -1  1   . -1 -1  3  3
X.17     3 -1  . -3  1  .  1 -1 -1  1  A   . -A  A -A   .  A -A  C -C
X.18     3 -1  . -3  1  .  1 -1 -1  1 -A   .  A -A  A   . -A  A -C  C
X.19     3  1  . -3 -1  .  1  1 -1 -1  A   . -A -A -A   .  A  A -C  C
X.20     3  1  . -3 -1  .  1  1 -1 -1 -A   .  A  A  A   . -A -A  C -C

A = -E(4)
  = -Sqrt(-1) = -i
B = 2*E(4)
  = 2*Sqrt(-1) = 2i
C = -3*E(4)
  = -3*Sqrt(-1) = -3i