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Properties

Label 16T600
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2^2\times D_4).C_2^3$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $600$
Group :  $(C_2^2\times D_4).C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,7,3,6)(2,8,4,5)(9,15,12,13)(10,16,11,14), (1,15)(2,16)(3,14)(4,13)(5,11,6,12)(7,10,8,9), (9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,14)(4,13)(5,6)(11,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $D_{4}$ x 12, $C_2^3$ x 15
16:  $D_4\times C_2$ x 18, $Q_8:C_2$ x 4, $C_2^4$
32:  $C_2^2 \wr C_2$ x 4, $C_2^3 : D_4 $ x 2, $C_2 \times (C_4\times C_2):C_2$ x 2, $C_2^2 \times D_4$ x 3
64:  16T73, 16T105, 16T115 x 4, 16T117
128:  32T1074

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

16T579 x 4, 16T600 x 7, 32T2846 x 2, 32T2847 x 2, 32T2848 x 8, 32T2849 x 4, 32T2850 x 2, 32T2945 x 4, 32T2946 x 4, 32T2947 x 4, 32T2948 x 4, 32T2949 x 4, 32T2950 x 4, 32T7246 x 2, 32T7271 x 2

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 16986]
Character table: Data not available.