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Properties

Label 25T3
Degree $25$
Order $50$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times D_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 3);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times D_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,5,6,4,10,3,9,2,8)(11,24,15,23,14,22,13,21,12,25)(16,18,20,17,19), (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $D_{5}$

Low degree siblings

10T6 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 $ $5$ $2$ $( 6,22)( 7,23)( 8,24)( 9,25)(10,21)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 2, 3, 4, 5)( 6,23, 8,25,10,22, 7,24, 9,21)(11,17,13,19,15,16,12,18,14,20)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 3, 5, 2, 4)( 6,24,10,23, 9,22, 8,21, 7,25)(11,18,15,17,14,16,13,20,12,19)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 4, 2, 5, 3)( 6,25, 7,21, 8,22, 9,23,10,24)(11,19,12,20,13,16,14,17,15,18)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 5, 4, 3, 2)( 6,21, 9,24, 7,22,10,25, 8,23)(11,20,14,18,12,16,15,19,13,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,12,18,25)( 2, 7,13,19,21)( 3, 8,14,20,22)( 4, 9,15,16,23) ( 5,10,11,17,24)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,14,16,24)( 2, 8,15,17,25)( 3, 9,11,18,21)( 4,10,12,19,22) ( 5, 6,13,20,23)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 8,11,19,23)( 2, 9,12,20,24)( 3,10,13,16,25)( 4, 6,14,17,21) ( 5, 7,15,18,22)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 9,13,17,22)( 2,10,14,18,23)( 3, 6,15,19,24)( 4, 7,11,20,25) ( 5, 8,12,16,21)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,10,15,20,21)( 2, 6,11,16,22)( 3, 7,12,17,23)( 4, 8,13,18,24) ( 5, 9,14,19,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,23, 8,19)( 2,12,24, 9,20)( 3,13,25,10,16)( 4,14,21, 6,17) ( 5,15,22, 7,18)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,25, 6,18)( 2,13,21, 7,19)( 3,14,22, 8,20)( 4,15,23, 9,16) ( 5,11,24,10,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,13,22, 9,17)( 2,14,23,10,18)( 3,15,24, 6,19)( 4,11,25, 7,20) ( 5,12,21, 8,16)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,14,24, 7,16)( 2,15,25, 8,17)( 3,11,21, 9,18)( 4,12,22,10,19) ( 5,13,23, 6,20)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,15,21,10,20)( 2,11,22, 6,16)( 3,12,23, 7,17)( 4,13,24, 8,18) ( 5,14,25, 9,19)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $50=2 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  50.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 5A1 5A-1 5A2 5A-2 5B1 5B2 5C1 5C-1 5C2 5C-2 5D1 5D-1 5D2 5D-2 10A1 10A-1 10A3 10A-3
Size 1 5 1 1 1 1 2 2 2 2 2 2 2 2 2 2 5 5 5 5
2 P 1A 1A 5A1 5A-1 5A-2 5A2 5D2 5B1 5C2 5D-1 5C-2 5B2 5C-1 5C1 5D-2 5D1 5A2 5A-2 5A-1 5A1
5 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A
Type
50.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.3.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
50.3.1c1 C 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
50.3.1c2 C 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
50.3.1c3 C 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
50.3.1c4 C 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
50.3.1d1 C 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
50.3.1d2 C 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
50.3.1d3 C 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
50.3.1d4 C 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
50.3.2a1 R 2 0 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 0 0 0 0
50.3.2a2 R 2 0 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 0 0 0 0
50.3.2b1 C 2 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 ζ5−1+ζ5 ζ5−2+ζ52 −1−ζ5−ζ52 ζ5+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 ζ5−1+1 1+ζ5 ζ5−2+1 1+ζ52 0 0 0 0
50.3.2b2 C 2 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5+ζ52 −1−ζ5−ζ52 −ζ5−2−1−ζ5 ζ5−2+ζ5 1+ζ5 ζ5−1+1 1+ζ52 ζ5−2+1 0 0 0 0
50.3.2b3 C 2 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−1−ζ5 ζ5−2+ζ5 −1−ζ5−ζ52 ζ5+ζ52 1+ζ52 ζ5−2+1 ζ5−1+1 1+ζ5 0 0 0 0
50.3.2b4 C 2 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ5 −ζ5−2−1−ζ5 ζ5+ζ52 −1−ζ5−ζ52 ζ5−2+1 1+ζ52 1+ζ5 ζ5−1+1 0 0 0 0
50.3.2c1 C 2 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 ζ5−2+ζ52 ζ5−1+ζ5 1+ζ52 ζ5−2+1 ζ5−1+1 1+ζ5 ζ5−2+ζ5 −ζ5−2−1−ζ5 ζ5+ζ52 −1−ζ5−ζ52 0 0 0 0
50.3.2c2 C 2 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+1 1+ζ52 1+ζ5 ζ5−1+1 −ζ5−2−1−ζ5 ζ5−2+ζ5 −1−ζ5−ζ52 ζ5+ζ52 0 0 0 0
50.3.2c3 C 2 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 ζ5−1+ζ5 ζ5−2+ζ52 1+ζ5 ζ5−1+1 1+ζ52 ζ5−2+1 −1−ζ5−ζ52 ζ5+ζ52 ζ5−2+ζ5 −ζ5−2−1−ζ5 0 0 0 0
50.3.2c4 C 2 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+1 1+ζ5 ζ5−2+1 1+ζ52 ζ5+ζ52 −1−ζ5−ζ52 −ζ5−2−1−ζ5 ζ5−2+ζ5 0 0 0 0

magma: CharacterTable(G);