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Properties

Label 25T46
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $D_5\times A_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 46);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times A_5$
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:  (1,2,3,4,5)(6,7,8,9,10)(11,22,18,14,25,16,12,23,19,15,21,17,13,24,20), (1,16,6,11,21)(2,20,7,15,22,5,17,10,12,25)(3,19,8,14,23,4,18,9,13,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$
$60$:  $A_5$
$120$:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $A_5$

Low degree siblings

30T128

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, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,18,15,17,14,16,13,20,12,19)(21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $75$ $2$ $( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25) (23,24)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 15, 5, 5 $ $40$ $15$ $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25)$
$ 15, 5, 5 $ $40$ $15$ $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,18,20,17,19)(21,23,25,22,24)$
$ 6, 6, 3, 2, 2, 2, 2, 1, 1 $ $100$ $6$ $( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(17,20)(18,19)(22,25)(23,24)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 6,11,21,16)( 2, 7,12,22,17)( 3, 8,13,23,18)( 4, 9,14,24,19) ( 5,10,15,25,20)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 7,13,24,20)( 2, 8,14,25,16)( 3, 9,15,21,17)( 4,10,11,22,18) ( 5, 6,12,23,19)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 8,15,22,19)( 2, 9,11,23,20)( 3,10,12,24,16)( 4, 6,13,25,17) ( 5, 7,14,21,18)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6,11,21,16)( 2,10,12,25,17, 5, 7,15,22,20)( 3, 9,13,24,18, 4, 8,14,23,19)$

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:  600.146
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 5B1 5B2 5C1 5C2 5D1 5D2 6A 10A1 10A3 10B1 10B3 15A1 15A2
Size 1 5 15 75 20 2 2 12 12 24 24 24 24 100 30 30 60 60 40 40
2 P 1A 1A 1A 1A 3A 5A2 5A1 5B2 5B1 5C1 5C2 5D1 5D2 3A 5A1 5A2 5B1 5B2 15A2 15A1
3 P 1A 2A 2B 2C 1A 5A2 5A1 5B2 5B1 5C1 5C2 5D1 5D2 2A 10A3 10A1 10B3 10B1 5A2 5A1
5 P 1A 2A 2B 2C 3A 1A 1A 1A 1A 1A 1A 1A 1A 6A 2B 2B 2A 2A 3A 3A
Type
600.146.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
600.146.1b R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1
600.146.2a1 R 2 0 2 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−2+ζ52 ζ5−1+ζ5
600.146.2a2 R 2 0 2 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−1+ζ5 ζ5−2+ζ52
600.146.3a1 R 3 3 −1 −1 0 3 3 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
600.146.3a2 R 3 3 −1 −1 0 3 3 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
600.146.3b1 R 3 −3 −1 1 0 3 3 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −1 −1 ζ5−2+ζ52 ζ5−1+ζ5 0 0
600.146.3b2 R 3 −3 −1 1 0 3 3 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −1 −1 ζ5−1+ζ5 ζ5−2+ζ52 0 0
600.146.4a R 4 4 0 0 1 4 4 −1 −1 −1 −1 −1 −1 1 0 0 −1 −1 1 1
600.146.4b R 4 −4 0 0 1 4 4 −1 −1 −1 −1 −1 −1 −1 0 0 1 1 1 1
600.146.5a R 5 5 1 1 −1 5 5 0 0 0 0 0 0 −1 1 1 0 0 −1 −1
600.146.5b R 5 −5 1 −1 −1 5 5 0 0 0 0 0 0 1 1 1 0 0 −1 −1
600.146.6a1 R 6 0 −2 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 1 1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
600.146.6a2 R 6 0 −2 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 1 1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
600.146.6b1 R 6 0 −2 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 1 1 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
600.146.6b2 R 6 0 −2 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 1 1 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
600.146.8a1 R 8 0 0 0 2 4ζ5−2+4ζ52 4ζ5−1+4ζ5 −2 −2 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
600.146.8a2 R 8 0 0 0 2 4ζ5−1+4ζ5 4ζ5−2+4ζ52 −2 −2 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
600.146.10a1 R 10 0 2 0 −2 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
600.146.10a2 R 10 0 2 0 −2 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52

magma: CharacterTable(G);