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Properties

Label 25T30
Degree $25$
Order $400$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_5^2.C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(25, 30);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5^2.C_2^2$
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:  (1,20,22,11,18,7,14,3,10,24)(2,15,23,6,19)(4,5,25,21,16,17,12,13,8,9), (1,7,25,19)(2,10,24,16)(3,8,23,18)(4,6,22,20)(5,9,21,17)(11,12,15,14), (1,15)(2,25)(3,10)(4,20)(6,13)(7,23)(9,18)(12,21)(14,16)(17,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T27 x 3, 20T90 x 3, 20T96 x 3, 20T97 x 3, 40T393 x 3, 40T394 x 3, 40T395 x 3

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 $ $10$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 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 $ $50$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2, 6, 5,21)( 3,11, 4,16)( 7,10,25,22)( 8,15,24,17)( 9,20,23,12)(13,14,19,18)$
$ 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 $ $8$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2,12,13,23,24, 9,10,20,16)( 3,22,14, 8,25,19, 6, 5,17,11)( 4, 7,15,18,21)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 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:  $400=2^{4} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  400.207
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 5A 5B 5C 10A 10B 10C
Size 1 10 10 10 25 25 25 50 50 50 8 8 8 40 40 40
2 P 1A 1A 1A 1A 1A 2D 2D 2D 2D 2D 5A 5B 5C 5A 5B 5C
5 P 1A 2C 2A 2B 2D 4A1 4A-1 4B 4C 4D 1A 1A 1A 2A 2B 2C
Type
400.207.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
400.207.1b R 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1
400.207.1c R 1 −1 −1 1 1 1 1 −1 1 −1 1 1 1 −1 −1 1
400.207.1d R 1 −1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 1 −1
400.207.1e R 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 1 1
400.207.1f R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 −1 −1
400.207.1g R 1 1 −1 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1
400.207.1h R 1 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1
400.207.2a1 C 2 0 0 0 −2 −2i 2i 0 0 0 2 2 2 0 0 0
400.207.2a2 C 2 0 0 0 −2 2i −2i 0 0 0 2 2 2 0 0 0
400.207.8a R 8 0 0 4 0 0 0 0 0 0 −2 −2 3 0 0 −1
400.207.8b R 8 0 4 0 0 0 0 0 0 0 −2 3 −2 0 −1 0
400.207.8c R 8 4 0 0 0 0 0 0 0 0 3 −2 −2 −1 0 0
400.207.8d R 8 −4 0 0 0 0 0 0 0 0 3 −2 −2 1 0 0
400.207.8e R 8 0 −4 0 0 0 0 0 0 0 −2 3 −2 0 1 0
400.207.8f R 8 0 0 −4 0 0 0 0 0 0 −2 −2 3 0 0 1

magma: CharacterTable(G);