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Properties

Label 15T26
Degree $15$
Order $405$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:C_5$

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Show commands: Magma

magma: G := TransitiveGroup(15, 26);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:C_5$
CHM label:   $[3^{4}]5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,6,11)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $C_5$

Low degree siblings

15T26 x 7, 45T61 x 2, 45T62 x 16, 45T63 x 8

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 4, 9,14)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 4,14, 9)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3,13, 8)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3,13, 8)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)$
$ 3, 3, 3, 3, 3 $ $5$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $405=3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  405.15
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E1 3E-1 3F1 3F-1 3G1 3G-1 3H1 3H-1 5A1 5A-1 5A2 5A-2
Size 1 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 81 81 81 81
3 P 1A 3H-1 3E-1 3A-1 3D-1 3B1 3A1 3G1 3C-1 3F-1 3F1 3E1 3G-1 3H1 3B-1 3D1 3C1 5A2 5A1 5A-2 5A-1
5 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 5A-2 5A-1 5A2 5A1
Type
405.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
405.15.1b1 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1
405.15.1b2 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5
405.15.1b3 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52
405.15.1b4 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2
405.15.5a1 C 5 −4−3ζ3 −1+3ζ3 −1 −1 2 2 2 2 2+3ζ3 −1−3ζ3 −1−3ζ3 2+3ζ3 −1 −1 −1 −1 0 0 0 0
405.15.5a2 C 5 −1+3ζ3 −4−3ζ3 −1 −1 2 2 2 2 −1−3ζ3 2+3ζ3 2+3ζ3 −1−3ζ3 −1 −1 −1 −1 0 0 0 0
405.15.5b1 C 5 −1 −1 −4−3ζ3 −1+3ζ3 −1 −1 −1 −1 −1−3ζ3 2+3ζ3 −1−3ζ3 2+3ζ3 2 2 2 2 0 0 0 0
405.15.5b2 C 5 −1 −1 −1+3ζ3 −4−3ζ3 −1 −1 −1 −1 2+3ζ3 −1−3ζ3 2+3ζ3 −1−3ζ3 2 2 2 2 0 0 0 0
405.15.5c1 C 5 −1 −1 2 2 −4−3ζ3 −1+3ζ3 −1−3ζ3 2+3ζ3 2 2 −1 −1 −1 −1 −1−3ζ3 2+3ζ3 0 0 0 0
405.15.5c2 C 5 −1 −1 2 2 −1+3ζ3 −4−3ζ3 2+3ζ3 −1−3ζ3 2 2 −1 −1 −1 −1 2+3ζ3 −1−3ζ3 0 0 0 0
405.15.5d1 C 5 −1 −1 2 2 −1−3ζ3 2+3ζ3 −1+3ζ3 −4−3ζ3 −1 −1 2 2 −1−3ζ3 2+3ζ3 −1 −1 0 0 0 0
405.15.5d2 C 5 −1 −1 2 2 2+3ζ3 −1−3ζ3 −4−3ζ3 −1+3ζ3 −1 −1 2 2 2+3ζ3 −1−3ζ3 −1 −1 0 0 0 0
405.15.5e1 C 5 −1−3ζ3 2+3ζ3 2+3ζ3 −1−3ζ3 −1 −1 2 2 −4−3ζ3 −1+3ζ3 −1 −1 2 2 −1 −1 0 0 0 0
405.15.5e2 C 5 2+3ζ3 −1−3ζ3 −1−3ζ3 2+3ζ3 −1 −1 2 2 −1+3ζ3 −4−3ζ3 −1 −1 2 2 −1 −1 0 0 0 0
405.15.5f1 C 5 −1−3ζ3 2+3ζ3 −1−3ζ3 2+3ζ3 2 2 −1 −1 −1 −1 −1+3ζ3 −4−3ζ3 −1 −1 2 2 0 0 0 0
405.15.5f2 C 5 2+3ζ3 −1−3ζ3 2+3ζ3 −1−3ζ3 2 2 −1 −1 −1 −1 −4−3ζ3 −1+3ζ3 −1 −1 2 2 0 0 0 0
405.15.5g1 C 5 2 2 −1 −1 −1 −1 −1−3ζ3 2+3ζ3 −1 −1 2 2 −4−3ζ3 −1+3ζ3 2+3ζ3 −1−3ζ3 0 0 0 0
405.15.5g2 C 5 2 2 −1 −1 −1 −1 2+3ζ3 −1−3ζ3 −1 −1 2 2 −1+3ζ3 −4−3ζ3 −1−3ζ3 2+3ζ3 0 0 0 0
405.15.5h1 C 5 2 2 −1 −1 −1−3ζ3 2+3ζ3 −1 −1 2 2 −1 −1 2+3ζ3 −1−3ζ3 −1+3ζ3 −4−3ζ3 0 0 0 0
405.15.5h2 C 5 2 2 −1 −1 2+3ζ3 −1−3ζ3 −1 −1 2 2 −1 −1 −1−3ζ3 2+3ζ3 −4−3ζ3 −1+3ζ3 0 0 0 0

magma: CharacterTable(G);