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Properties

Label 16T732
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^3.Q_8.C_6$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $732$
Group :  $C_2^3.Q_8.C_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,4)(5,12,15,7,9,14,6,11,16,8,10,13), (1,10,8)(2,9,7)(3,12,5)(4,11,6)(13,14)(15,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$, $\SL(2,3)$ x 2
48:  16T59
96:  12T60
192:  24T293

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 8: $\SL(2,3)$

Low degree siblings

16T732, 32T9339

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

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 615]
Character table: Data not available.