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Properties

Label 15T19
Degree $15$
Order $300$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_5^2 : C_4):C_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 19);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_5^2 : C_4):C_3$
CHM label:   $[5^{2}:4]3$
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:  (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T19, 25T26, 30T78 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{15}$ $1$ $1$ $()$
$2^{6},1^{3}$ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$
$4^{3},1^{3}$ $25$ $4$ $( 3, 9, 6,15)( 4, 7,13,10)( 5, 8,14,11)$
$4^{3},1^{3}$ $25$ $4$ $( 3,15, 6, 9)( 4,10,13, 7)( 5,11,14, 8)$
$5^{2},1^{5}$ $12$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$3^{5}$ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$6^{2},3$ $25$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$
$12,3$ $25$ $12$ $( 1, 2, 3, 7,14,12,10, 5, 9, 4, 8,15)( 6,13,11)$
$12,3$ $25$ $12$ $( 1, 2, 3,13, 8, 6, 7, 5,12,10,14, 9)( 4,11,15)$
$3^{5}$ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$
$6^{2},3$ $25$ $6$ $( 1, 3, 5,13, 6, 2)( 4,15, 8,10, 9,14)( 7,12,11)$
$12,3$ $25$ $12$ $( 1, 3, 8,13,12,11, 4, 9, 5, 7,15, 2)( 6,14,10)$
$12,3$ $25$ $12$ $( 1, 3,14, 7, 6, 8, 4,12,11,13, 9, 2)( 5,10,15)$
$5^{3}$ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  300.24
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 5A 5B 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 25 25 25 25 25 12 12 25 25 25 25 25 25
2 P 1A 1A 3A-1 3A1 2A 2A 5A 5B 3A1 3A-1 6A-1 6A-1 6A1 6A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 5B 2A 2A 4A1 4A-1 4A-1 4A1
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 1A 6A-1 6A1 12A1 12A-5 12A-1 12A5
Type
300.24.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
300.24.1b R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
300.24.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
300.24.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
300.24.1d1 C 1 −1 1 1 −i i 1 1 −1 −1 i −i i −i
300.24.1d2 C 1 −1 1 1 i −i 1 1 −1 −1 −i i −i i
300.24.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
300.24.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
300.24.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125
300.24.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12
300.24.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125
300.24.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12
300.24.12a R 12 0 0 0 0 0 −3 2 0 0 0 0 0 0
300.24.12b R 12 0 0 0 0 0 2 −3 0 0 0 0 0 0

magma: CharacterTable(G);