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Properties

Label 25T21
Degree $25$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_5\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(25, 21);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\wr C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,3,24,5,21,2,23,4,25)(6,17,8,19,10,16,7,18,9,20)(11,12,13,14,15), (1,16,17,7,8,23,24,14,15,5)(2,6,18,22,9,13,25,4,11,20)(3,21,19,12,10)
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 5: None

Low degree siblings

10T19, 10T21 x 2, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 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

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 $ $10$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2,11)( 3,21)( 4, 6)( 5,16)( 7,14)( 8,24)(10,19)(13,22)(15,17)(18,25)$
$ 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 $ $20$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 2,12,13,23,24, 9,10,20,16)( 3,22,14, 8,25,19, 6, 5,17,11)( 4, 7,15,18,21)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 3,23,25,20,17,12,14, 9, 6)( 2,13,24,10,16)( 4, 8,21, 5,18,22,15,19, 7,11)$
$ 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 $ $4$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$

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:  200.43
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 5B1 5B2 5C 10A1 10A3 10B1 10B3
Size 1 10 10 25 50 4 4 4 4 8 20 20 20 20
2 P 1A 1A 1A 1A 2C 5A1 5B2 5A2 5B1 5C 5A1 5B1 5B2 5A2
5 P 1A 2A 2B 2C 4A 1A 1A 1A 1A 1A 2A 2B 2B 2A
Type
200.43.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.43.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
200.43.1c R 1 −1 1 1 −1 1 1 1 1 1 −1 −1 1 1
200.43.1d R 1 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1
200.43.2a R 2 0 0 −2 0 2 2 2 2 2 0 0 0 0
200.43.4a1 R 4 0 2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 ζ5−2+ζ52 ζ5−1+ζ5
200.43.4a2 R 4 0 2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 ζ5−1+ζ5 ζ5−2+ζ52
200.43.4b1 R 4 2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−1+ζ5 ζ5−2+ζ52 0 0
200.43.4b2 R 4 2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 ζ5−2+ζ52 ζ5−1+ζ5 0 0
200.43.4c1 R 4 −2 0 0 0 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
200.43.4c2 R 4 −2 0 0 0 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
200.43.4d1 R 4 0 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
200.43.4d2 R 4 0 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
200.43.8a R 8 0 0 0 0 −2 −2 −2 −2 3 0 0 0 0

magma: CharacterTable(G);