# Properties

 Label 10T17 Degree $10$ Order $200$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $(C_5^2 : C_4) : C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 17);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $17$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $(C_5^2 : C_4) : C_2$ CHM label: $[5^{2}:4]2$ 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: (1,7,9,3)(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
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$ x 2
$40$:  $F_{5}\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: None

## Low degree siblings

10T17, 20T54 x 2, 25T19, 40T169 x 2

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

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: 200.42 magma: IdentifyGroup(G);
 Character table:  2 3 3 3 3 . 3 3 3 3 1 1 1 . 1 5 2 . . . 2 1 . . 1 1 1 2 2 2 1a 4a 4b 2a 5a 2b 4c 4d 2c 10a 10b 5b 5c 5d 2P 1a 2a 2a 1a 5a 1a 2a 2a 1a 5b 5d 5b 5c 5d 3P 1a 4b 4a 2a 5a 2b 4d 4c 2c 10a 10b 5b 5c 5d 5P 1a 4a 4b 2a 1a 2b 4c 4d 2c 2b 2c 1a 1a 1a 7P 1a 4b 4a 2a 5a 2b 4d 4c 2c 10a 10b 5b 5c 5d 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 1 A -A -1 1 -1 -A A 1 -1 1 1 1 1 X.6 1 -A A -1 1 -1 A -A 1 -1 1 1 1 1 X.7 1 A -A -1 1 1 A -A -1 1 -1 1 1 1 X.8 1 -A A -1 1 1 -A A -1 1 -1 1 1 1 X.9 4 . . . -1 . . . -4 . 1 4 -1 -1 X.10 4 . . . -1 . . . 4 . -1 4 -1 -1 X.11 4 . . . -1 -4 . . . 1 . -1 -1 4 X.12 4 . . . -1 4 . . . -1 . -1 -1 4 X.13 8 . . . -2 . . . . . . -2 3 -2 X.14 8 . . . 3 . . . . . . -2 -2 -2 A = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);