Properties

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

Related objects

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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 1, 1$ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$ $2, 2, 2, 2, 1, 1$ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$ $5, 1, 1, 1, 1, 1$ $8$ $5$ $( 2, 4, 6, 8,10)$ $4, 4, 2$ $50$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$ $4, 4, 2$ $50$ $4$ $( 1, 2)( 3, 6, 9, 8)( 4, 5,10, 7)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $5, 5$ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$

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);