↑

Properties

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

Downloads

Learn more

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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{25}$ $1$ $1$ $0$ $()$
2A $2^{12},1$ $25$ $2$ $12$ $( 1,12)( 2,11)( 3,15)( 4,14)( 5,13)( 6, 7)( 8,10)(16,22)(17,21)(18,25)(19,24)(20,23)$
3A $3^{8},1$ $50$ $3$ $16$ $( 1,13, 3)( 2, 7, 8)( 4,25,18)( 5,19,23)( 6,14,22)( 9,21,12)(10,20,17)(11,15,16)$
4A1 $4^{6},1$ $25$ $4$ $18$ $( 1,20,12,23)( 2,18,11,25)( 3,16,15,22)( 4,19,14,24)( 5,17,13,21)( 6,10, 7, 8)$
4A-1 $4^{6},1$ $25$ $4$ $18$ $( 1,23,12,20)( 2,25,11,18)( 3,22,15,16)( 4,24,14,19)( 5,21,13,17)( 6, 8, 7,10)$
5A $5^{5}$ $24$ $5$ $20$ $( 1,23,20,12, 9)( 2,24,16,13,10)( 3,25,17,14, 6)( 4,21,18,15, 7)( 5,22,19,11, 8)$
6A $6^{4},1$ $50$ $6$ $20$ $( 1, 5,24,19,20,21)( 2, 6, 4,18,14,16)( 3,12, 9,17, 8,11)( 7,10,23,13,15,22)$
8A1 $8^{3},1$ $75$ $8$ $21$ $( 1,10,20, 7,12, 8,23, 6)( 2,16,18,15,11,22,25, 3)( 4,13,19,21,14, 5,24,17)$
8A-1 $8^{3},1$ $75$ $8$ $21$ $( 1, 8,20, 6,12,10,23, 7)( 2,22,18, 3,11,16,25,15)( 4, 5,19,17,14,13,24,21)$
8A3 $8^{3},1$ $75$ $8$ $21$ $( 1, 6,23, 8,12, 7,20,10)( 2, 3,25,22,11,15,18,16)( 4,17,24, 5,14,21,19,13)$
8A-3 $8^{3},1$ $75$ $8$ $21$ $( 1, 7,23,10,12, 6,20, 8)( 2,15,25,16,11, 3,18,22)( 4,21,24,13,14,17,19, 5)$
12A1 $12^{2},1$ $50$ $12$ $22$ $( 1,17, 6,19,12, 9,13,22, 8,25, 2,10)( 3,23,21,20, 5, 4,11,16,18,24,14,15)$
12A-1 $12^{2},1$ $50$ $12$ $22$ $( 1,21,23,17, 7, 6,24, 4, 2, 8,18,19)( 3,20,13,16,25,11,22,10,12, 9, 5,14)$

Malle's constant $a(G)$:     $1/12$

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 4A1 4A1 4A-1 4A-1 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);