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Properties

Label 25T45
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_5^2:C_3:C_8$

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Show commands: Magma

magma: G := TransitiveGroup(25, 45);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:C_3:C_8$
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:  (1,8,21,12,6,4,11,25)(2,19,24,20,10,18,13,17)(3,5,22,23,9,7,15,14), (1,14,20,24,12,4,23,19)(2,7,18,8,11,6,25,10)(3,5,16,17,15,13,22,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$8$:  $C_8$
$12$:  $C_3 : C_4$
$24$:  24T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

30T143

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 2, 2, 2, 2, 2, 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 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $50$ $3$ $( 2,25, 6)( 3,19,11)( 4,13,16)( 5, 7,21)( 8,20,10)( 9,14,15)(12,22,24) (17,23,18)$
$ 6, 6, 6, 6, 1 $ $50$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
$ 12, 12, 1 $ $50$ $12$ $( 2,13,21, 3,25,16, 5,19, 6, 4, 7,11)( 8,23,22,15,20,18,24, 9,10,17,12,14)$
$ 12, 12, 1 $ $50$ $12$ $( 2,19,21, 4,25,11, 5,13, 6, 3, 7,16)( 8, 9,22,17,20,14,24,23,10,15,12,18)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,12, 4, 9, 5,20, 3,23)( 6,22,16,14,21,10,11,18)( 7, 8,19,17,25,24,13,15)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,20, 4,23, 5,12, 3, 9)( 6,10,16,18,21,22,11,14)( 7,24,19,15,25, 8,13,17)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2, 9, 3,12, 5,23, 4,20)( 6,14,11,22,21,18,16,10)( 7,17,13, 8,25,15,19,24)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,23, 3,20, 5, 9, 4,12)( 6,18,11,10,21,14,16,22)( 7,15,13,24,25,17,19, 8)$

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:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  600.148
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 4A1 4A-1 5A 6A 8A1 8A-1 8A3 8A-3 12A1 12A-1
Size 1 25 50 25 25 24 50 75 75 75 75 50 50
2 P 1A 1A 3A 2A 2A 5A 3A 4A-1 4A-1 4A1 4A1 6A 6A
3 P 1A 2A 1A 4A-1 4A1 5A 2A 8A1 8A-3 8A-1 8A3 4A1 4A-1
5 P 1A 2A 3A 4A1 4A-1 1A 6A 8A-1 8A3 8A1 8A-3 12A1 12A-1
Type
600.148.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
600.148.1b R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
600.148.1c1 C 1 1 1 −1 −1 1 1 −i i i −i −1 −1
600.148.1c2 C 1 1 1 −1 −1 1 1 i −i −i i −1 −1
600.148.1d1 C 1 −1 1 −ζ82 ζ82 1 −1 ζ83 −ζ8 ζ8 −ζ83 ζ82 −ζ82
600.148.1d2 C 1 −1 1 ζ82 −ζ82 1 −1 −ζ8 ζ83 −ζ83 ζ8 −ζ82 ζ82
600.148.1d3 C 1 −1 1 −ζ82 ζ82 1 −1 −ζ83 ζ8 −ζ8 ζ83 ζ82 −ζ82
600.148.1d4 C 1 −1 1 ζ82 −ζ82 1 −1 ζ8 −ζ83 ζ83 −ζ8 −ζ82 ζ82
600.148.2a R 2 2 −1 2 2 2 −1 0 0 0 0 −1 −1
600.148.2b S 2 2 −1 −2 −2 2 −1 0 0 0 0 1 1
600.148.2c1 C 2 −2 −1 −2i 2i 2 1 0 0 0 0 −i i
600.148.2c2 C 2 −2 −1 2i −2i 2 1 0 0 0 0 i −i
600.148.24a R 24 0 0 0 0 −1 0 0 0 0 0 0 0

magma: CharacterTable(G);