↑

Properties

Label 20T106
Order \(400\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{10}^2:C_2^2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $106$
Group :  $C_{10}^2:C_2^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(2,8)(3,5)(4,6)(11,14)(12,13)(15,20)(16,19), (1,17,8,13,4,19,9,15,6,11,2,18,7,14,3,20,10,16,5,12), (1,4,6,7,10)(2,3,5,8,9)(11,18,14,20,16,12,17,13,19,15)
$|\Aut(F/K)|$:  $2$

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, $C_2^3$
10:  $D_{5}$ x 2
16:  $D_4\times C_2$
20:  $D_{10}$ x 6
40:  20T8 x 2
80:  20T21 x 2
100:  $D_5^2$
200:  20T59

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: None

Degree 10: $D_5^2$

Low degree siblings

20T106 x 7, 40T318 x 4, 40T388 x 4, 40T389 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $10$ $(11,13,16,18,19,12,14,15,17,20)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $(11,14,16,17,19)(12,13,15,18,20)$
$ 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $10$ $(11,15,19,13,17,12,16,20,14,18)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $(11,16,19,14,17)(12,15,20,13,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,20)(14,19)(15,18)(16,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $50$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12)(13,19)(14,20)(15,17)(16,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,13,16,18,19,12,14,15,17,20)$
$ 5, 5, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,14,16,17,19)(12,13,15,18,20)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,15,19,13,17,12,16,20,14,18)$
$ 5, 5, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,16,19,14,17)(12,15,20,13,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $25$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 7)( 6, 8)(11,12)(13,19)(14,20)(15,17)(16,18)$
$ 10, 10 $ $2$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,13,16,18,19,12,14,15,17,20)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,14,16,17,19)(12,13,15,18,20)$
$ 10, 10 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,15,19,13,17,12,16,20,14,18)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,16,19,14,17)(12,15,20,13,18)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,17,14,19,16)(12,18,13,20,15)$
$ 10, 10 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,18,14,20,16,12,17,13,19,15)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,19,17,16,14)(12,20,18,15,13)$
$ 10, 10 $ $2$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,20,17,15,14,12,19,18,16,13)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,14,16,17,19)(12,13,15,18,20)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,15,19,13,17,12,16,20,14,18)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,16,19,14,17)(12,15,20,13,18)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,17,14,19,16)(12,18,13,20,15)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,18,14,20,16,12,17,13,19,15)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,19,17,16,14)(12,20,18,15,13)$
$ 10, 10 $ $2$ $10$ $( 1, 5,10, 3, 7, 2, 6, 9, 4, 8)(11,15,19,13,17,12,16,20,14,18)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 5,10, 3, 7, 2, 6, 9, 4, 8)(11,16,19,14,17)(12,15,20,13,18)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 5,10, 3, 7, 2, 6, 9, 4, 8)(11,17,14,19,16)(12,18,13,20,15)$
$ 10, 10 $ $2$ $10$ $( 1, 5,10, 3, 7, 2, 6, 9, 4, 8)(11,18,14,20,16,12,17,13,19,15)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,10, 4, 7)( 2, 5, 9, 3, 8)(11,16,19,14,17)(12,15,20,13,18)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,10, 4, 7)( 2, 5, 9, 3, 8)(11,17,14,19,16)(12,18,13,20,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,20)(10,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$
$ 20 $ $20$ $20$ $( 1,11, 3,13, 6,16, 8,18,10,19, 2,12, 4,14, 5,15, 7,17, 9,20)$
$ 10, 10 $ $20$ $10$ $( 1,11, 4,14, 6,16, 7,17,10,19)( 2,12, 3,13, 5,15, 8,18, 9,20)$
$ 20 $ $20$ $20$ $( 1,11, 5,15,10,19, 3,13, 7,17, 2,12, 6,16, 9,20, 4,14, 8,18)$
$ 10, 10 $ $20$ $10$ $( 1,11, 6,16,10,19, 4,14, 7,17)( 2,12, 5,15, 9,20, 3,13, 8,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,11)( 2,12)( 3,20)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,13)(10,14)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 2,12)( 3,20, 4,19)( 5,18, 6,17)( 7,16, 8,15)( 9,13,10,14)$
$ 20 $ $20$ $20$ $( 1,11, 3,20, 6,17, 8,15,10,14, 2,12, 4,19, 5,18, 7,16, 9,13)$
$ 10, 10 $ $20$ $10$ $( 1,11, 4,19, 6,17, 7,16,10,14)( 2,12, 3,20, 5,18, 8,15, 9,13)$
$ 20 $ $20$ $20$ $( 1,11, 5,18,10,14, 3,20, 7,16, 2,12, 6,17, 9,13, 4,19, 8,15)$
$ 10, 10 $ $20$ $10$ $( 1,11, 6,17,10,14, 4,19, 7,16)( 2,12, 5,18, 9,13, 3,20, 8,15)$

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 180]
Character table: Data not available.