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Properties

Label 16T764
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^3:S_4.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $764$
Group :  $C_2^3:S_4.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,15,4,10,13,2,12,16,3,9,14)(5,7,6,8), (1,16,9,5)(2,15,10,6)(3,14,12,8)(4,13,11,7)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$ x 3
48:  $S_4\times C_2$ x 3
96:  $V_4^2:S_3$
192:  12T100

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

16T764 x 5, 32T9367 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 $ $6$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 5, 9,15)( 6,10,16)( 7,12,14)( 8,11,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $24$ $2$ $( 3, 4)( 7, 8)( 9,15)(10,16)(11,14)(12,13)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3, 4)( 5, 6)( 9,15,10,16)(11,14,12,13)$
$ 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 $ $32$ $6$ $( 1, 2)( 3, 4)( 5, 9,15, 6,10,16)( 7,12,14, 8,11,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 12, 4 $ $32$ $12$ $( 1, 3, 2, 4)( 5,11,16, 8,10,14, 6,12,15, 7, 9,13)$
$ 12, 4 $ $32$ $12$ $( 1, 3, 2, 4)( 5,13,10, 8,16,12, 6,14, 9, 7,15,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,16)(10,15)(11,14)(12,13)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 5, 9,15)( 2, 6,10,16)( 3, 8,12,13)( 4, 7,11,14)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 5, 9,16)( 2, 6,10,15)( 3, 8,12,14)( 4, 7,11,13)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 7,10,14)( 2, 8, 9,13)( 3, 5,11,15)( 4, 6,12,16)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 7,10,13)( 2, 8, 9,14)( 3, 5,11,16)( 4, 6,12,15)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,14,10,13)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,16)(12,15)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 20097]
Character table: Data not available.