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Properties

Label 25T2
Degree $25$
Order $25$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_5^2$

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Show commands: Magma

magma: G := TransitiveGroup(25, 2);
 

Group action invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$ x 6

Low degree siblings

There are no siblings 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 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 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 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,10,14,18,22)( 2, 6,15,19,23)( 3, 7,11,20,24)( 4, 8,12,16,25) ( 5, 9,13,17,21)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,11,21, 6,16)( 2,12,22, 7,17)( 3,13,23, 8,18)( 4,14,24, 9,19) ( 5,15,25,10,20)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,13,25, 7,19)( 2,14,21, 8,20)( 3,15,22, 9,16)( 4,11,23,10,17) ( 5,12,24, 6,18)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,14,22,10,18)( 2,15,23, 6,19)( 3,11,24, 7,20)( 4,12,25, 8,16) ( 5,13,21, 9,17)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,15,24, 8,17)( 2,11,25, 9,18)( 3,12,21,10,19)( 4,13,22, 6,20) ( 5,14,23, 7,16)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,16, 6,21,11)( 2,17, 7,22,12)( 3,18, 8,23,13)( 4,19, 9,24,14) ( 5,20,10,25,15)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,17, 8,24,15)( 2,18, 9,25,11)( 3,19,10,21,12)( 4,20, 6,22,13) ( 5,16, 7,23,14)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,18,10,22,14)( 2,19, 6,23,15)( 3,20, 7,24,11)( 4,16, 8,25,12) ( 5,17, 9,21,13)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,19, 7,25,13)( 2,20, 8,21,14)( 3,16, 9,22,15)( 4,17,10,23,11) ( 5,18, 6,24,12)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,20, 9,23,12)( 2,16,10,24,13)( 3,17, 6,25,14)( 4,18, 7,21,15) ( 5,19, 8,22,11)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,21,16,11, 6)( 2,22,17,12, 7)( 3,23,18,13, 8)( 4,24,19,14, 9) ( 5,25,20,15,10)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,22,18,14,10)( 2,23,19,15, 6)( 3,24,20,11, 7)( 4,25,16,12, 8) ( 5,21,17,13, 9)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,23,20,12, 9)( 2,24,16,13,10)( 3,25,17,14, 6)( 4,21,18,15, 7) ( 5,22,19,11, 8)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,25,19,13, 7)( 2,21,20,14, 8)( 3,22,16,15, 9)( 4,23,17,11,10) ( 5,24,18,12, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $25=5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  25.2
magma: IdentifyGroup(G);
 
Character table:

1A 5A1 5A-1 5A2 5A-2 5B1 5B-1 5B2 5B-2 5C1 5C-1 5C2 5C-2 5D1 5D-1 5D2 5D-2 5E1 5E-1 5E2 5E-2 5F1 5F-1 5F2 5F-2
Size 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 P 1A 5C-2 5D-2 5E-1 5C-1 5B2 5E-2 5D1 5E2 5C1 5A-2 5C2 5F2 5B1 5D-1 5E1 5F-2 5B-2 5A2 5A1 5D2 5B-1 5F-1 5A-1 5F1
Type
25.2.1a R 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
25.2.1b1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1
25.2.1b2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1
25.2.1b3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1
25.2.1b4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1
25.2.1c1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5
25.2.1c2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1
25.2.1c3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2
25.2.1c4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52
25.2.1d1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52
25.2.1d2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2
25.2.1d3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5
25.2.1d4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1
25.2.1e1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
25.2.1e2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
25.2.1e3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
25.2.1e4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
25.2.1f1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2
25.2.1f2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52
25.2.1f3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1
25.2.1f4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5
25.2.1g1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
25.2.1g2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52
25.2.1g3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
25.2.1g4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5

magma: CharacterTable(G);