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Properties

Label 20T105
Order \(400\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:D_5.Q_8$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $105$
Group :  $C_5:D_5.Q_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,7,19,16)(4,8,20,15)(5,10,17,14)(6,9,18,13)(11,12), (1,11,14,3,5,16,18,8,9,20,2,12,13,4,6,15,17,7,10,19)
$|\Aut(F/K)|$:  $2$

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:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $C_4:C_4$
200:  $D_5^2 : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $D_5^2 : C_2$

Low degree siblings

20T105, 40T336 x 2, 40T338 x 2, 40T363, 40T365

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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $50$ $4$ $( 5, 9,17,13)( 6,10,18,14)( 7,16,20,11)( 8,15,19,12)$
$ 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 $ $8$ $5$ $( 3, 8,12,15,19)( 4, 7,11,16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $50$ $4$ $( 1, 2)( 3, 4)( 5,10,17,14)( 6, 9,18,13)( 7,15,20,12)( 8,16,19,11)$
$ 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 $ $8$ $10$ $( 1, 2)( 3, 7,12,16,19, 4, 8,11,15,20)( 5, 6)( 9,10)(13,14)(17,18)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,12, 6,11)( 7,13, 8,14)( 9,19,10,20)(15,18,16,17)$
$ 20 $ $20$ $20$ $( 1, 3, 6, 7, 9,12,14,16,17,19, 2, 4, 5, 8,10,11,13,15,18,20)$
$ 20 $ $20$ $20$ $( 1, 3, 6,11, 9,19,14, 7,17,15, 2, 4, 5,12,10,20,13, 8,18,16)$
$ 20 $ $20$ $20$ $( 1, 3, 6,16, 9, 8,14,20,17,12, 2, 4, 5,15,10, 7,13,19,18,11)$
$ 20 $ $20$ $20$ $( 1, 3, 6,20, 9,15,14,11,17, 8, 2, 4, 5,19,10,16,13,12,18, 7)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7,14, 8,13)( 9,20,10,19)(15,17,16,18)$
$ 20 $ $20$ $20$ $( 1, 4, 6, 8, 9,11,14,15,17,20, 2, 3, 5, 7,10,12,13,16,18,19)$
$ 20 $ $20$ $20$ $( 1, 4, 6,12, 9,20,14, 8,17,16, 2, 3, 5,11,10,19,13, 7,18,15)$
$ 20 $ $20$ $20$ $( 1, 4, 6,15, 9, 7,14,19,17,11, 2, 3, 5,16,10, 8,13,20,18,12)$
$ 20 $ $20$ $20$ $( 1, 4, 6,19, 9,16,14,12,17, 7, 2, 3, 5,20,10,15,13,11,18, 8)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 8,12,15,19)( 4, 7,11,16,20)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3,12,19, 8,15)( 4,11,20, 7,16)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3,15, 8,19,12)( 4,16, 7,20,11)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3,19,15,12, 8)( 4,20,16,11, 7)$
$ 10, 10 $ $4$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 7,12,16,19, 4, 8,11,15,20)$
$ 10, 10 $ $4$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3,11,19, 7,15, 4,12,20, 8,16)$
$ 10, 10 $ $4$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3,16, 8,20,12, 4,15, 7,19,11)$
$ 10, 10 $ $4$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3,20,15,11, 8, 4,19,16,12, 7)$

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 130]
Character table: Data not available.