# Properties

 Label 10T19 Degree $10$ Order $200$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_5^2 : C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 19);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $19$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_5^2 : C_2$ CHM label: $[5^{2}:4_{2}]2$ 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)(2,7)(3,8)(4,9)(5,10) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: None

## Low degree siblings

10T21 x 2, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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

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: 200.43 magma: IdentifyGroup(G);
 Character table:  2 3 2 3 . 2 2 1 1 1 1 1 1 1 1 5 2 . . 2 1 1 1 1 1 1 2 2 2 2 1a 4a 2a 5a 2b 2c 10a 10b 10c 10d 5b 5c 5d 5e 2P 1a 2a 1a 5a 1a 1a 5b 5c 5d 5e 5e 5d 5c 5b 3P 1a 4a 2a 5a 2b 2c 10d 10c 10b 10a 5e 5d 5c 5b 5P 1a 4a 2a 1a 2b 2c 2b 2c 2c 2b 1a 1a 1a 1a 7P 1a 4a 2a 5a 2b 2c 10d 10c 10b 10a 5e 5d 5c 5b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 -1 1 1 -1 1 1 1 1 X.3 1 -1 1 1 1 -1 1 -1 -1 1 1 1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 X.5 2 . -2 2 . . . . . . 2 2 2 2 X.6 4 . . -1 -2 . A . . *A B *C C *B X.7 4 . . -1 -2 . *A . . A *B C *C B X.8 4 . . -1 . -2 . A *A . C B *B *C X.9 4 . . -1 . -2 . *A A . *C *B B C X.10 4 . . -1 . 2 . -*A -A . *C *B B C X.11 4 . . -1 . 2 . -A -*A . C B *B *C X.12 4 . . -1 2 . -*A . . -A *B C *C B X.13 4 . . -1 2 . -A . . -*A B *C C *B X.14 8 . . 3 . . . . . . -2 -2 -2 -2 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-Sqrt(5))/2 = 1-b5 C = 2*E(5)+2*E(5)^4 = -1+Sqrt(5) = 2b5 

magma: CharacterTable(G);