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Properties

Label 25T19
Degree $25$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 19);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5:F_5$
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:  (1,22,16,12,6,2,21,17,11,7)(3,25,18,15,8,5,23,20,13,10)(4,24,19,14,9), (1,12,19,8)(2,14,18,6)(3,11,17,9)(4,13,16,7)(5,15,20,10)(21,22,24,23)
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 5: $F_5$ x 2

Low degree siblings

10T17 x 2, 20T54 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $5$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

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);