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Properties

Label 16T754
Order \(384\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $Q_8:C_2^2.D_6$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $754$
Group :  $Q_8:C_2^2.D_6$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,2,11)(3,9,4,10)(5,14,6,13)(7,15,8,16), (1,3)(2,4)(5,14,9,7,15,12)(6,13,10,8,16,11), (1,6,15,10,2,5,16,9)(3,7,14,12,4,8,13,11)
$|\Aut(F/K)|$:  $2$

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

16T738, 16T750 x 2, 32T9342, 32T9357

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

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 20092]
Character table: Data not available.