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Properties

Label 15T13
Degree $15$
Order $150$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_5^2 : C_3):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(15, 13);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_5^2 : C_3):C_2$
CHM label:   $[5^{2}]S(3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,10,7,4)(2,5,8,11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T14, 25T16, 30T37, 30T38

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 10, 5 $ $15$ $10$ $( 1, 2, 4, 5, 7, 8,10,11,13,14)( 3,15,12, 9, 6)$
$ 10, 5 $ $15$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,12, 6,15, 9)$
$ 10, 5 $ $15$ $10$ $( 1, 2,10,11, 4, 5,13,14, 7, 8)( 3, 9,15, 6,12)$
$ 10, 5 $ $15$ $10$ $( 1, 2,13,14,10,11, 7, 8, 4, 5)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $3$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,15,12, 9, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  150.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A-1 5A2 5A-2 5B1 5B2 10A1 10A-1 10A3 10A-3
Size 1 15 50 3 3 3 3 6 6 15 15 15 15
2 P 1A 1A 3A 5A-2 5A1 5A-1 5A2 5B2 5B1 5A2 5A1 5A-2 5A-1
3 P 1A 2A 1A 5A2 5A-1 5A1 5A-2 5B2 5B1 10A1 10A3 10A-1 10A-3
5 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A
Type
150.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
150.5.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
150.5.2a R 2 0 −1 2 2 2 2 2 2 0 0 0 0
150.5.3a1 C 3 1 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ52 ζ5−2 ζ5 ζ5−1
150.5.3a2 C 3 1 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2 ζ52 ζ5−1 ζ5
150.5.3a3 C 3 1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5 ζ5−1 ζ5−2 ζ52
150.5.3a4 C 3 1 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1 ζ5 ζ52 ζ5−2
150.5.3b1 C 3 −1 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1
150.5.3b2 C 3 −1 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
150.5.3b3 C 3 −1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52
150.5.3b4 C 3 −1 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
150.5.6a1 R 6 0 0 −2ζ5−1−2ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −2ζ5−2−2ζ52 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0 0 0
150.5.6a2 R 6 0 0 −2ζ5−2−2ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −2ζ5−1−2ζ5 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 0 0 0

magma: CharacterTable(G);