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Properties

Label 25T44
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_5:S_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 44);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:S_5$
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:  (1,20,6,5,16,10)(2,19,7,4,17,9)(3,18,8)(11,23)(12,22)(13,21)(14,25)(15,24), (1,24,2,25,3,21,4,22,5,23)(6,9,7,10,8)(11,16,12,17,13,18,14,19,15,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $S_5$

Low degree siblings

30T144

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 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $50$ $2$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,10)( 7, 9)(12,15)(13,14)(21,25)(22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)$
$ 10, 10, 5 $ $30$ $10$ $( 1,17, 3,19, 5,16, 2,18, 4,20)( 6,22, 8,24,10,21, 7,23, 9,25)(11,12,13,14,15)$
$ 10, 10, 5 $ $30$ $10$ $( 1,18, 5,17, 4,16, 3,20, 2,19)( 6,23,10,22, 9,21, 8,25, 7,24)(11,13,15,12,14)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,16,21)( 2,17,22)( 3,18,23)( 4,19,24)( 5,20,25)$
$ 15, 5, 5 $ $40$ $15$ $( 1,17,23, 4,20,21, 2,18,24, 5,16,22, 3,19,25)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 15, 5, 5 $ $40$ $15$ $( 1,18,25, 2,19,21, 3,20,22, 4,16,23, 5,17,24)( 6, 8,10, 7, 9)(11,13,15,12,14)$
$ 6, 6, 3, 2, 2, 2, 2, 2 $ $100$ $6$ $( 1,20,21, 5,16,25)( 2,19,22, 4,17,24)( 3,18,23)( 6,13)( 7,12)( 8,11)( 9,15) (10,14)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $150$ $4$ $( 1,20,21,10)( 2,19,22, 9)( 3,18,23, 8)( 4,17,24, 7)( 5,16,25, 6)(12,15) (13,14)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,16,21, 6,14)( 2,17,22, 7,15)( 3,18,23, 8,11)( 4,19,24, 9,12) ( 5,20,25,10,13)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,17,23, 9,13)( 2,18,24,10,14)( 3,19,25, 6,15)( 4,20,21, 7,11) ( 5,16,22, 8,12)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,18,25, 7,12)( 2,19,21, 8,13)( 3,20,22, 9,14)( 4,16,23,10,15) ( 5,17,24, 6,11)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,20,24, 8,15)( 2,16,25, 9,11)( 3,17,21,10,12)( 4,18,22, 6,13) ( 5,19,23, 7,14)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,19,22,10,11)( 2,20,23, 6,12)( 3,16,24, 7,13)( 4,17,25, 8,14) ( 5,18,21, 9,15)$

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:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  600.145
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A1 5A2 5B 5C1 5C-1 5C2 5C-2 6A 10A1 10A3 15A1 15A2
Size 1 15 50 20 150 2 2 24 24 24 24 24 100 30 30 40 40
2 P 1A 1A 1A 3A 2A 5A2 5A1 5C-1 5C-2 5B 5C1 5C2 3A 5A2 5A1 15A2 15A1
3 P 1A 2A 2B 1A 4A 5A2 5A1 5C1 5C2 5B 5C-1 5C-2 2B 10A3 10A1 5A1 5A2
5 P 1A 2A 2B 3A 4A 1A 1A 1A 1A 1A 1A 1A 6A 2A 2A 3A 3A
Type
600.145.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
600.145.1b R 1 1 −1 1 −1 1 1 1 1 1 1 1 −1 1 1 1 1
600.145.2a1 R 2 2 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
600.145.2a2 R 2 2 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
600.145.4a R 4 0 2 1 0 4 4 −1 −1 −1 −1 −1 −1 0 0 1 1
600.145.4b R 4 0 −2 1 0 4 4 −1 −1 −1 −1 −1 1 0 0 1 1
600.145.5a R 5 1 −1 −1 1 5 5 0 0 0 0 0 −1 1 1 −1 −1
600.145.5b R 5 1 1 −1 −1 5 5 0 0 0 0 0 1 1 1 −1 −1
600.145.6a R 6 −2 0 0 0 6 6 1 1 1 1 1 0 −2 −2 0 0
600.145.6b1 C 6 −2 0 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 1 −ζ5−2−1−2ζ5 2ζ5−2+1+2ζ5+ζ52 −1−ζ5−2ζ52 −ζ5−2+ζ5+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
600.145.6b2 C 6 −2 0 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 1 2ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5 −ζ5−2+ζ5+ζ52 −1−ζ5−2ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
600.145.6b3 C 6 −2 0 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 1 −1−ζ5−2ζ52 −ζ5−2+ζ5+ζ52 2ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
600.145.6b4 C 6 −2 0 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 1 −ζ5−2+ζ5+ζ52 −1−ζ5−2ζ52 −ζ5−2−1−2ζ5 2ζ5−2+1+2ζ5+ζ52 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
600.145.8a1 R 8 0 0 2 0 4ζ5−2+4ζ52 4ζ5−1+4ζ5 −2 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
600.145.8a2 R 8 0 0 2 0 4ζ5−1+4ζ5 4ζ5−2+4ζ52 −2 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
600.145.10a1 R 10 2 0 −2 0 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
600.145.10a2 R 10 2 0 −2 0 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5

magma: CharacterTable(G);