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Properties

Label 16T573
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $573$
Group :  $C_2^4.C_2^4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,12,2,11)(3,14,4,13)(5,15,6,16)(7,10,8,9), (1,2)(3,4)(5,6)(7,8), (1,2)(3,8)(4,7)(5,6)(9,13)(10,14), (1,8)(2,7)(3,5)(4,6)
$|\Aut(F/K)|$:  $4$

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 20, $C_2^3$ x 15
16:  $D_4\times C_2$ x 30, $C_2^4$
32:  $C_2^2 \wr C_2$ x 8, $C_2^3 : D_4 $ x 2, $C_2^2 \times D_4$ x 5
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T87, 16T105 x 2, 16T109 x 4
128:  16T245 x 2, 32T1237

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_2^2 \wr C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T531 x 32, 16T540 x 32, 16T573 x 31, 32T2636 x 64, 32T2637 x 64, 32T2638 x 16, 32T2639 x 16, 32T2640 x 8, 32T2641 x 16, 32T2680 x 8, 32T2681 x 8, 32T2682 x 8, 32T2683 x 16, 32T2684 x 16, 32T2685 x 8, 32T2827 x 16, 32T2828 x 8, 32T2829 x 8, 32T2830 x 16, 32T4757 x 16, 32T6815 x 4

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 16888]
Character table: Data not available.