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Properties

Label 16T330
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $330$
Group :  $C_2^4.D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,4,6,7,9,11,14,15)(2,3,5,8,10,12,13,16), (1,4,2,3)(5,16,13,8)(6,15,14,7)(9,11,10,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
32:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
64:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $QD_{16}$, $(((C_4 \times C_2): C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T330 x 3, 16T339 x 4, 32T750, 32T751 x 2, 32T752 x 2, 32T776, 32T1574, 32T1576, 32T1682 x 2, 32T1751 x 2, 32T1790

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 75]
Character table: Data not available.