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Properties

Label 25T47
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_5^2:C_{24}$

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Show commands: Magma

magma: G := TransitiveGroup(25, 47);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:C_{24}$
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:  (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), (1,17,10,16,21,9,7,14,20,12,22,18,19,3,15,4,24,11,13,6,5,8,23,2)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$8$:  $C_8$
$12$:  $C_{12}$
$24$:  $C_{24}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

30T131

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

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:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  600.149
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 5A 6A1 6A-1 8A1 8A-1 8A3 8A-3 12A1 12A-1 12A5 12A-5 24A1 24A-1 24A5 24A-5 24A7 24A-7 24A11 24A-11
Size 1 25 25 25 25 25 24 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
2 P 1A 1A 3A-1 3A1 2A 2A 5A 3A1 3A-1 4A1 4A-1 4A-1 4A1 6A1 6A-1 6A-1 6A1 12A1 12A5 12A-1 12A1 12A-5 12A5 12A-5 12A-1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 2A 2A 8A3 8A-3 8A1 8A-1 4A-1 4A1 4A-1 4A1 8A1 8A1 8A-1 8A-3 8A-1 8A-3 8A3 8A3
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 6A-1 6A1 8A-3 8A3 8A-1 8A1 12A-1 12A1 12A-5 12A5 24A5 24A-11 24A-5 24A-7 24A11 24A1 24A-1 24A7
Type
600.149.1a 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
600.149.1b 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
600.149.1c1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
600.149.1c2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
600.149.1d1 C 1 1 1 1 −1 −1 1 1 1 −i i i −i −1 −1 −1 −1 i −i i −i −i i −i i
600.149.1d2 C 1 1 1 1 −1 −1 1 1 1 i −i −i i −1 −1 −1 −1 −i i −i i i −i i −i
600.149.1e1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 −1 −1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
600.149.1e2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 −1 −1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
600.149.1f1 C 1 −1 1 1 −ζ82 ζ82 1 −1 −1 ζ83 −ζ8 ζ8 −ζ83 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ83 −ζ8 ζ83 −ζ83 ζ8 ζ83 −ζ8
600.149.1f2 C 1 −1 1 1 ζ82 −ζ82 1 −1 −1 −ζ8 ζ83 −ζ83 ζ8 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ8 ζ83 −ζ8 ζ8 −ζ83 −ζ8 ζ83
600.149.1f3 C 1 −1 1 1 −ζ82 ζ82 1 −1 −1 −ζ83 ζ8 −ζ8 ζ83 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ83 ζ8 −ζ83 ζ83 −ζ8 −ζ83 ζ8
600.149.1f4 C 1 −1 1 1 ζ82 −ζ82 1 −1 −1 ζ8 −ζ83 ζ83 −ζ8 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ8 −ζ83 ζ8 −ζ8 ζ83 ζ8 −ζ83
600.149.1g1 C 1 1 −ζ122 ζ124 −1 −1 1 ζ124 −ζ122 −ζ123 ζ123 ζ123 −ζ123 ζ122 −ζ124 −ζ124 ζ122 −ζ12 ζ125 −ζ125 ζ12 ζ12 −ζ125 ζ125 −ζ12
600.149.1g2 C 1 1 ζ124 −ζ122 −1 −1 1 −ζ122 ζ124 ζ123 −ζ123 −ζ123 ζ123 −ζ124 ζ122 ζ122 −ζ124 ζ125 −ζ12 ζ12 −ζ125 −ζ125 ζ12 −ζ12 ζ125
600.149.1g3 C 1 1 −ζ122 ζ124 −1 −1 1 ζ124 −ζ122 ζ123 −ζ123 −ζ123 ζ123 ζ122 −ζ124 −ζ124 ζ122 ζ12 −ζ125 ζ125 −ζ12 −ζ12 ζ125 −ζ125 ζ12
600.149.1g4 C 1 1 ζ124 −ζ122 −1 −1 1 −ζ122 ζ124 −ζ123 ζ123 ζ123 −ζ123 −ζ124 ζ122 ζ122 −ζ124 −ζ125 ζ12 −ζ12 ζ125 ζ125 −ζ12 ζ12 −ζ125
600.149.1h1 C 1 −1 −ζ244 ζ248 −ζ246 ζ246 1 −ζ248 ζ244 ζ249 −ζ243 ζ243 −ζ249 −ζ2410 ζ242 −ζ242 ζ2410 ζ2411 −ζ24 ζ247 −ζ245 ζ245 −ζ247 ζ24 −ζ2411
600.149.1h2 C 1 −1 ζ248 −ζ244 ζ246 −ζ246 1 ζ244 −ζ248 −ζ243 ζ249 −ζ249 ζ243 ζ242 −ζ2410 ζ2410 −ζ242 −ζ24 ζ2411 −ζ245 ζ247 −ζ247 ζ245 −ζ2411 ζ24
600.149.1h3 C 1 −1 −ζ244 ζ248 −ζ246 ζ246 1 −ζ248 ζ244 −ζ249 ζ243 −ζ243 ζ249 −ζ2410 ζ242 −ζ242 ζ2410 −ζ2411 ζ24 −ζ247 ζ245 −ζ245 ζ247 −ζ24 ζ2411
600.149.1h4 C 1 −1 ζ248 −ζ244 ζ246 −ζ246 1 ζ244 −ζ248 ζ243 −ζ249 ζ249 −ζ243 ζ242 −ζ2410 ζ2410 −ζ242 ζ24 −ζ2411 ζ245 −ζ247 ζ247 −ζ245 ζ2411 −ζ24
600.149.1h5 C 1 −1 −ζ244 ζ248 ζ246 −ζ246 1 −ζ248 ζ244 −ζ243 ζ249 −ζ249 ζ243 ζ2410 −ζ242 ζ242 −ζ2410 ζ245 −ζ247 ζ24 −ζ2411 ζ2411 −ζ24 ζ247 −ζ245
600.149.1h6 C 1 −1 ζ248 −ζ244 −ζ246 ζ246 1 ζ244 −ζ248 ζ249 −ζ243 ζ243 −ζ249 −ζ242 ζ2410 −ζ2410 ζ242 −ζ247 ζ245 −ζ2411 ζ24 −ζ24 ζ2411 −ζ245 ζ247
600.149.1h7 C 1 −1 −ζ244 ζ248 ζ246 −ζ246 1 −ζ248 ζ244 ζ243 −ζ249 ζ249 −ζ243 ζ2410 −ζ242 ζ242 −ζ2410 −ζ245 ζ247 −ζ24 ζ2411 −ζ2411 ζ24 −ζ247 ζ245
600.149.1h8 C 1 −1 ζ248 −ζ244 −ζ246 ζ246 1 ζ244 −ζ248 −ζ249 ζ243 −ζ243 ζ249 −ζ242 ζ2410 −ζ2410 ζ242 ζ247 −ζ245 ζ2411 −ζ24 ζ24 −ζ2411 ζ245 −ζ247
600.149.24a R 24 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);