Properties

Label 10T20
Order \(200\)
n \(10\)
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 $n$ :  $10$
Transitive number $t$ :  $20$
Group :  $C_5^2 : Q_8$
CHM label :  $[5^{2}:4_{2}]2_{2}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,9,7)(2,4,8,6), (2,4,6,8,10), (1,6,9,4)(2,3,8,7)(5,10)
$|\Aut(F/K)|$:  $1$

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: $C_2$

Degree 5: None

Low degree siblings

10T20 x 2, 20T47 x 3, 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 $ $1$ $1$ $()$
$ 4, 4, 1, 1 $ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 4, 4, 2 $ $50$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$
$ 4, 4, 2 $ $50$ $4$ $( 1, 2)( 3, 6, 9, 8)( 4, 5,10, 7)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$

Group invariants

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

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