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Properties

Label 25T4
Degree $25$
Order $50$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{25}$

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Show commands: Magma

magma: G := TransitiveGroup(25, 4);
 

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $50=2 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  50.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 5A1 5A2 25A1 25A2 25A3 25A4 25A6 25A7 25A8 25A9 25A11 25A12
Size 1 25 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 5A2 5A1 25A3 25A8 25A11 25A4 25A1 25A7 25A9 25A12 25A2 25A6
5 P 1A 2A 1A 1A 5A1 5A1 5A2 5A2 5A2 5A1 5A2 5A1 5A1 5A2
Type
50.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
50.1.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
50.1.2b1 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−12+ζ2512 ζ25−1+ζ25 ζ25−11+ζ2511 ζ25−2+ζ252 ζ25−3+ζ253 ζ25−9+ζ259 ζ25−4+ζ254 ζ25−8+ζ258 ζ25−7+ζ257 ζ25−6+ζ256
50.1.2b2 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−8+ζ258 ζ25−9+ζ259 ζ25−1+ζ25 ζ25−7+ζ257 ζ25−2+ζ252 ζ25−6+ζ256 ζ25−11+ζ2511 ζ25−3+ζ253 ζ25−12+ζ2512 ζ25−4+ζ254
50.1.2b3 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−7+ζ257 ζ25−11+ζ2511 ζ25−4+ζ254 ζ25−3+ζ253 ζ25−8+ζ258 ζ25−1+ζ25 ζ25−6+ζ256 ζ25−12+ζ2512 ζ25−2+ζ252 ζ25−9+ζ259
50.1.2b4 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−3+ζ253 ζ25−6+ζ256 ζ25−9+ζ259 ζ25−12+ζ2512 ζ25−7+ζ257 ζ25−4+ζ254 ζ25−1+ζ25 ζ25−2+ζ252 ζ25−8+ζ258 ζ25−11+ζ2511
50.1.2b5 R 2 0 ζ25−10+ζ2510 ζ25−5+ζ255 ζ25−2+ζ252 ζ25−4+ζ254 ζ25−6+ζ256 ζ25−8+ζ258 ζ25−12+ζ2512 ζ25−11+ζ2511 ζ25−9+ζ259 ζ25−7+ζ257 ζ25−3+ζ253 ζ25−1+ζ25
50.1.2b6 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−11+ζ2511 ζ25−3+ζ253 ζ25−8+ζ258 ζ25−6+ζ256 ζ25−9+ζ259 ζ25−2+ζ252 ζ25−12+ζ2512 ζ25−1+ζ25 ζ25−4+ζ254 ζ25−7+ζ257
50.1.2b7 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−9+ζ259 ζ25−7+ζ257 ζ25−2+ζ252 ζ25−11+ζ2511 ζ25−4+ζ254 ζ25−12+ζ2512 ζ25−3+ζ253 ζ25−6+ζ256 ζ25−1+ζ25 ζ25−8+ζ258
50.1.2b8 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−6+ζ256 ζ25−12+ζ2512 ζ25−7+ζ257 ζ25−1+ζ25 ζ25−11+ζ2511 ζ25−8+ζ258 ζ25−2+ζ252 ζ25−4+ζ254 ζ25−9+ζ259 ζ25−3+ζ253
50.1.2b9 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−4+ζ254 ζ25−8+ζ258 ζ25−12+ζ2512 ζ25−9+ζ259 ζ25−1+ζ25 ζ25−3+ζ253 ζ25−7+ζ257 ζ25−11+ζ2511 ζ25−6+ζ256 ζ25−2+ζ252
50.1.2b10 R 2 0 ζ25−5+ζ255 ζ25−10+ζ2510 ζ25−1+ζ25 ζ25−2+ζ252 ζ25−3+ζ253 ζ25−4+ζ254 ζ25−6+ζ256 ζ25−7+ζ257 ζ25−8+ζ258 ζ25−9+ζ259 ζ25−11+ζ2511 ζ25−12+ζ2512

magma: CharacterTable(G);