Properties

Label 20T47
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2:Q_8$

Related objects

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Group action invariants

Degree $n$:  $20$
Transitive number $t$:  $47$
Group:  $C_5^2:Q_8$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,9,14,6)(2,10,13,5)(3,20,12,15)(4,19,11,16)(7,8)(17,18), (1,16,10,3)(2,15,9,4)(5,19)(6,20)(7,17,12,14)(8,18,11,13)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $C_5^2 : Q_8$

Low degree siblings

10T20 x 3, 20T47 x 2, 25T17, 40T166 x 3

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

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [200, 44]
Character table:   
     2  3  3  .  2  2  2  .  .
     5  2  .  2  .  .  .  2  2

       1a 2a 5a 4a 4b 4c 5b 5c
    2P 1a 1a 5a 2a 2a 2a 5b 5c
    3P 1a 2a 5a 4a 4b 4c 5b 5c
    5P 1a 2a 1a 4a 4b 4c 1a 1a

X.1     1  1  1  1  1  1  1  1
X.2     1  1  1 -1 -1  1  1  1
X.3     1  1  1 -1  1 -1  1  1
X.4     1  1  1  1 -1 -1  1  1
X.5     2 -2  2  .  .  .  2  2
X.6     8  .  3  .  .  . -2 -2
X.7     8  . -2  .  .  . -2  3
X.8     8  . -2  .  .  .  3 -2