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Properties

Label 16T747
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^3:S_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $747$
Group :  $C_2\times C_2^3:S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13)(2,14)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16), (1,2)(3,14,9,15)(4,13,10,16)(5,7)(6,8)(11,12), (1,10,16,7)(2,9,15,8)(3,14,5,11)(4,13,6,12)
$|\Aut(F/K)|$:  $4$

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:  $S_4$ x 3
48:  $S_4\times C_2$ x 3
96:  $V_4^2:S_3$
192:  $C_2^3:S_4$ x 2, 12T100

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4\times C_2$, $C_2^3:S_4$ x 2

Low degree siblings

16T747 x 11, 32T9354 x 3

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 $ $6$ $2$ $( 5, 8)( 6, 7)(13,16)(14,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 5,13)( 6,14)( 7,15)( 8,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3, 5, 9, 8)( 4, 6,10, 7)(13,16)(14,15)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 3, 5,13)( 4, 6,14)( 7,15,10)( 8,16, 9)$
$ 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 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 4)( 5,14)( 6,13)( 7,16)( 8,15)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3, 6, 9, 7)( 4, 5,10, 8)(11,12)(13,15)(14,16)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 2)( 3, 6,13, 4, 5,14)( 7,16,10, 8,15, 9)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 3, 5,12, 9, 8)( 2, 4, 6,11,10, 7)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 3, 5,13)( 2, 4, 6,14)( 7,15,11,10)( 8,16,12, 9)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 3, 5,16)( 2, 4, 6,15)( 7,14,11,10)( 8,13,12, 9)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 3,12, 9)( 2, 4,11,10)( 5,13, 8,16)( 6,14, 7,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 3)( 5,14)( 6,13)( 7,16)( 8,15)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 4, 5,11, 9, 7)( 2, 3, 6,12,10, 8)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 4, 5,14)( 2, 3, 6,13)( 7,16,11, 9)( 8,15,12,10)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 4, 5,15)( 2, 3, 6,16)( 7,13,11, 9)( 8,14,12,10)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 4,12,10)( 2, 3,11, 9)( 5,14, 8,15)( 6,13, 7,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,16)(14,15)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 20089]
Character table: Data not available.