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Properties

Label 20T100
Order \(400\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_5\wr C_2$

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $100$
Group :  $C_2\times D_5\wr C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,4)(5,18)(6,17)(7,8)(9,14)(10,13)(11,12)(15,16)(19,20), (1,20,6,16)(2,19,5,15)(3,9,11,18)(4,10,12,17)(7,13)(8,14), (1,12,13,19)(2,11,14,20)(3,5,7,10)(4,6,8,9)(15,18)(16,17)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
200:  $D_5^2 : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $D_5^2 : C_2$

Low degree siblings

20T92 x 4, 20T98 x 2, 20T100 x 3, 40T329 x 4, 40T340 x 2, 40T342 x 2, 40T362, 40T366, 40T373 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, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7,20)( 8,19)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $25$ $2$ $( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3, 7,11,16,20)( 4, 8,12,15,19)$
$ 5, 5, 2, 2, 2, 2, 1, 1 $ $20$ $10$ $( 3, 7,11,16,20)( 4, 8,12,15,19)( 5,17)( 6,18)( 9,13)(10,14)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3,11,20, 7,16)( 4,12,19, 8,15)$
$ 5, 5, 2, 2, 2, 2, 1, 1 $ $20$ $10$ $( 3,11,20, 7,16)( 4,12,19, 8,15)( 5,17)( 6,18)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,10)(11,15)(12,16)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $25$ $2$ $( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,19)( 8,20)( 9,14)(10,13)(11,15)(12,16)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 8,11,15,20, 4, 7,12,16,19)( 5, 6)( 9,10)(13,14)(17,18)$
$ 10, 2, 2, 2, 2, 2 $ $20$ $10$ $( 1, 2)( 3, 8,11,15,20, 4, 7,12,16,19)( 5,18)( 6,17)( 9,14)(10,13)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3,12,20, 8,16, 4,11,19, 7,15)( 5, 6)( 9,10)(13,14)(17,18)$
$ 10, 2, 2, 2, 2, 2 $ $20$ $10$ $( 1, 2)( 3,12,20, 8,16, 4,11,19, 7,15)( 5,18)( 6,17)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)$
$ 4, 4, 4, 4, 2, 2 $ $50$ $4$ $( 1, 3)( 2, 4)( 5, 8,17,19)( 6, 7,18,20)( 9,11,13,16)(10,12,14,15)$
$ 10, 10 $ $20$ $10$ $( 1, 3, 6, 7, 9,11,13,16,18,20)( 2, 4, 5, 8,10,12,14,15,17,19)$
$ 10, 10 $ $20$ $10$ $( 1, 3, 9,11,18,20, 6, 7,13,16)( 2, 4,10,12,17,19, 5, 8,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)$
$ 4, 4, 4, 4, 2, 2 $ $50$ $4$ $( 1, 4)( 2, 3)( 5, 7,17,20)( 6, 8,18,19)( 9,12,13,15)(10,11,14,16)$
$ 10, 10 $ $20$ $10$ $( 1, 4, 6, 8, 9,12,13,15,18,19)( 2, 3, 5, 7,10,11,14,16,17,20)$
$ 10, 10 $ $20$ $10$ $( 1, 4, 9,12,18,19, 6, 8,13,15)( 2, 3,10,11,17,20, 5, 7,14,16)$
$ 10, 10 $ $4$ $10$ $( 1, 5, 9,14,18, 2, 6,10,13,17)( 3, 8,11,15,20, 4, 7,12,16,19)$
$ 10, 10 $ $8$ $10$ $( 1, 5, 9,14,18, 2, 6,10,13,17)( 3,12,20, 8,16, 4,11,19, 7,15)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 6, 9,13,18)( 2, 5,10,14,17)( 3, 7,11,16,20)( 4, 8,12,15,19)$
$ 5, 5, 5, 5 $ $8$ $5$ $( 1, 6, 9,13,18)( 2, 5,10,14,17)( 3,11,20, 7,16)( 4,12,19, 8,15)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 9,18, 6,13)( 2,10,17, 5,14)( 3,11,20, 7,16)( 4,12,19, 8,15)$
$ 10, 10 $ $4$ $10$ $( 1,10,18, 5,13, 2, 9,17, 6,14)( 3,12,20, 8,16, 4,11,19, 7,15)$

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 211]
Character table: Data not available.