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Properties

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

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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);
 
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

LabelCycle TypeSizeOrderRepresentative
$ 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);
 
Nilpotency class:   not nilpotent
Label:  200.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A 5B 5C 5D 10A 10B
Size 1 5 5 25 25 25 25 25 4 4 8 8 20 20
2 P 1A 1A 1A 1A 2C 2C 2C 2C 5A 5B 5C 5D 5A 5B
5 P 1A 2A 2B 2C 4B-1 4B1 4A-1 4A1 1A 1A 1A 1A 2A 2B
Type
200.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.42.1b R 1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1
200.42.1c R 1 −1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1
200.42.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
200.42.1e1 C 1 −1 1 −1 −i i −i i 1 1 1 1 −1 1
200.42.1e2 C 1 −1 1 −1 i −i i −i 1 1 1 1 −1 1
200.42.1f1 C 1 1 −1 −1 −i i i −i 1 1 1 1 1 −1
200.42.1f2 C 1 1 −1 −1 i −i −i i 1 1 1 1 1 −1
200.42.4a R 4 0 4 0 0 0 0 0 4 −1 −1 −1 0 −1
200.42.4b R 4 4 0 0 0 0 0 0 −1 4 −1 −1 −1 0
200.42.4c R 4 −4 0 0 0 0 0 0 −1 4 −1 −1 1 0
200.42.4d R 4 0 −4 0 0 0 0 0 4 −1 −1 −1 0 1
200.42.8a R 8 0 0 0 0 0 0 0 −2 −2 −2 3 0 0
200.42.8b R 8 0 0 0 0 0 0 0 −2 −2 3 −2 0 0

magma: CharacterTable(G);