↑

Properties

Label 25T6
Degree $25$
Order $75$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_5^2:C_3$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(25, 6);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T9 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $25$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $25$ $3$ $( 2,25, 6)( 3,19,11)( 4,13,16)( 5, 7,21)( 8,20,10)( 9,14,15)(12,22,24) (17,23,18)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1,17, 8,24,15)( 2,18, 9,25,11)( 3,19,10,21,12)( 4,20, 6,22,13) ( 5,16, 7,23,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $75=3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  75.2
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 5A1 5A-1 5A2 5A-2 5B1 5B-1 5B2 5B-2
Size 1 25 25 3 3 3 3 3 3 3 3
3 P 1A 3A-1 3A1 5A2 5B1 5A-1 5B-1 5A-2 5B-2 5B2 5A1
5 P 1A 1A 1A 5A-2 5B-1 5A1 5B1 5A2 5B2 5B-2 5A-1
Type
75.2.1a R 1 1 1 1 1 1 1 1 1 1 1
75.2.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
75.2.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1
75.2.3a1 C 3 0 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
75.2.3a2 C 3 0 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
75.2.3a3 C 3 0 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
75.2.3a4 C 3 0 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
75.2.3b1 C 3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52
75.2.3b2 C 3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52
75.2.3b3 C 3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52
75.2.3b4 C 3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5

magma: CharacterTable(G);