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Properties

Label 10T20
Degree $10$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2 : Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(10, 20);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2 : Q_8$
CHM label:  $[5^{2}:4_{2}]2_{2}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,9,7)(2,4,8,6), (2,4,6,8,10), (1,6,9,4)(2,3,8,7)(5,10)
magma: Generators(G);
 

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 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  200.44
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);