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Properties

Label 25T42
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_5\times S_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 42);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times S_5$
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:  (1,16,15,5,20,14,4,19,13,3,18,12,2,17,11)(6,23,7,24,8,25,9,21,10,22), (1,25,12,9,19)(2,21,13,10,20)(3,22,14,6,16)(4,23,15,7,17)(5,24,11,8,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $S_5$

Low degree siblings

30T145

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle 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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1,18)( 2,19)( 3,20)( 4,16)( 5,17)$
$ 10, 5, 5, 5 $ $10$ $10$ $( 1,20, 5,19, 4,18, 3,17, 2,16)( 6, 8,10, 7, 9)(11,13,15,12,14) (21,23,25,22,24)$
$ 10, 5, 5, 5 $ $10$ $10$ $( 1,17, 4,20, 2,18, 5,16, 3,19)( 6,10, 9, 8, 7)(11,15,14,13,12) (21,25,24,23,22)$
$ 10, 5, 5, 5 $ $10$ $10$ $( 1,16, 2,17, 3,18, 4,19, 5,20)( 6, 9, 7,10, 8)(11,14,12,15,13) (21,24,22,25,23)$
$ 10, 5, 5, 5 $ $10$ $10$ $( 1,19, 3,16, 5,18, 2,20, 4,17)( 6, 7, 8, 9,10)(11,12,13,14,15) (21,22,23,24,25)$
$ 10, 10, 5 $ $15$ $10$ $( 1,17, 4,20, 2,18, 5,16, 3,19)( 6,10, 9, 8, 7)(11,22,14,25,12,23,15,21,13,24)$
$ 10, 10, 5 $ $15$ $10$ $( 1,19, 3,16, 5,18, 2,20, 4,17)( 6, 7, 8, 9,10)(11,24,13,21,15,23,12,25,14,22)$
$ 10, 10, 5 $ $15$ $10$ $( 1,16, 2,17, 3,18, 4,19, 5,20)( 6, 9, 7,10, 8)(11,21,12,22,13,23,14,24,15,25)$
$ 10, 10, 5 $ $15$ $10$ $( 1,20, 5,19, 4,18, 3,17, 2,16)( 6, 8,10, 7, 9)(11,25,15,24,14,23,13,22,12,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1,18)( 2,19)( 3,20)( 4,16)( 5,17)(11,23)(12,24)(13,25)(14,21)(15,22)$
$ 15, 5, 5 $ $20$ $15$ $( 1,16,22, 5,20,21, 4,19,25, 3,18,24, 2,17,23)( 6, 9, 7,10, 8)(11,14,12,15,13)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,18,21)( 2,19,22)( 3,20,23)( 4,16,24)( 5,17,25)$
$ 15, 5, 5 $ $20$ $15$ $( 1,20,25, 2,16,21, 3,17,22, 4,18,23, 5,19,24)( 6, 8,10, 7, 9)(11,13,15,12,14)$
$ 15, 5, 5 $ $20$ $15$ $( 1,19,23, 4,17,21, 2,20,24, 5,18,22, 3,16,25)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 15, 5, 5 $ $20$ $15$ $( 1,17,24, 3,19,21, 5,16,23, 2,18,25, 4,20,22)( 6,10, 9, 8, 7)(11,15,14,13,12)$
$ 15, 10 $ $20$ $30$ $( 1,16,22, 5,20,21, 4,19,25, 3,18,24, 2,17,23)( 6,11, 7,12, 8,13, 9,14,10,15)$
$ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2 $ $20$ $6$ $( 1,18,21)( 2,19,22)( 3,20,23)( 4,16,24)( 5,17,25)( 6,13)( 7,14)( 8,15)( 9,11) (10,12)$
$ 15, 10 $ $20$ $30$ $( 1,20,25, 2,16,21, 3,17,22, 4,18,23, 5,19,24)( 6,15,10,14, 9,13, 8,12, 7,11)$
$ 15, 10 $ $20$ $30$ $( 1,19,23, 4,17,21, 2,20,24, 5,18,22, 3,16,25)( 6,14, 8,11,10,13, 7,15, 9,12)$
$ 15, 10 $ $20$ $30$ $( 1,17,24, 3,19,21, 5,16,23, 2,18,25, 4,20,22)( 6,12, 9,15, 7,13,10,11, 8,14)$
$ 20, 5 $ $30$ $20$ $( 1,17,24,11, 2,18,25,12, 3,19,21,13, 4,20,22,14, 5,16,23,15)( 6,10, 9, 8, 7)$
$ 20, 5 $ $30$ $20$ $( 1,19,23,12, 5,18,22,11, 4,17,21,15, 3,16,25,14, 2,20,24,13)( 6, 7, 8, 9,10)$
$ 20, 5 $ $30$ $20$ $( 1,16,22,13, 3,18,24,15, 5,20,21,12, 2,17,23,14, 4,19,25,11)( 6, 9, 7,10, 8)$
$ 20, 5 $ $30$ $20$ $( 1,20,25,15, 4,18,23,13, 2,16,21,11, 5,19,24,14, 3,17,22,12)( 6, 8,10, 7, 9)$
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ $30$ $4$ $( 1,18,21,14)( 2,19,22,15)( 3,20,23,11)( 4,16,24,12)( 5,17,25,13)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,17,24,11, 8)( 2,18,25,12, 9)( 3,19,21,13,10)( 4,20,22,14, 6) ( 5,16,23,15, 7)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,19,23,12, 6)( 2,20,24,13, 7)( 3,16,25,14, 8)( 4,17,21,15, 9) ( 5,18,22,11,10)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,16,22,13, 9)( 2,17,23,14,10)( 3,18,24,15, 6)( 4,19,25,11, 7) ( 5,20,21,12, 8)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,20,25,15,10)( 2,16,21,11, 6)( 3,17,22,12, 7)( 4,18,23,13, 8) ( 5,19,24,14, 9)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,18,21,14, 7)( 2,19,22,15, 8)( 3,20,23,11, 9)( 4,16,24,12,10) ( 5,17,25,13, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  600.144
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);