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Properties

Label 16T225
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2.C_2^2:D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $225$
Group :  $C_2^2.C_2^2:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,12,3,14,2,11,4,13)(5,16,7,9,6,15,8,10), (1,4,2,3)(5,8,6,7), (1,2)(3,4)(9,11,10,12)(13,16,14,15)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
64:  32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_2^2 \wr C_2$

Low degree siblings

16T225 x 3, 16T311 x 2, 32T500 x 2, 32T501 x 2, 32T502 x 2, 32T720, 32T721, 32T1960, 32T1998

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 638]
Character table: Data not available.