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Properties

Label 15T34
Degree $15$
Order $810$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:D_5$

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Show commands: Magma

magma: G := TransitiveGroup(15, 34);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:D_5$
CHM label:   $[3^{4}]D(5)$
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:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T34 x 3, 15T35 x 4, 30T191 x 4, 30T192 x 4, 45T121 x 8, 45T122 x 8, 45T123, 45T124

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 $ $10$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)( 4, 9,14)( 5,15,10)$
$ 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 $ $5$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 5, 5, 5 $ $162$ $5$ $( 1,12, 3, 4,10)( 2, 8, 9,15, 6)( 5,11, 7,13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1, 6,11)( 2,15)( 3,14,13, 9, 8, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$
$ 6, 6, 3 $ $45$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15,12,10, 7, 5)( 3, 9, 8,14,13, 4)$
$ 6, 6, 3 $ $45$ $6$ $( 1,11, 6)( 2,15,12,10, 7, 5)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1, 6,11)( 2,15,12,10, 7, 5)( 3, 4)( 8, 9)(13,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $810=2 \cdot 3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  810.101
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E 3F 3G 3H 5A1 5A2 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1
Size 1 45 5 5 5 5 5 5 5 5 10 10 10 10 162 162 45 45 45 45 45 45 45 45
2 P 1A 1A 3A-1 3C1 3B-1 3A1 3C-1 3B1 3D-1 3D1 3H 3F 3E 3G 5A2 5A1 3B1 3B-1 3C1 3A-1 3A1 3D1 3C-1 3D-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 5A2 5A1 2A 2A 2A 2A 2A 2A 2A 2A
5 P 1A 2A 3A-1 3C1 3B-1 3A1 3C-1 3B1 3D-1 3D1 3H 3F 3E 3G 1A 1A 6B-1 6B1 6C-1 6A1 6A-1 6D-1 6C1 6D1
Type
810.101.1a 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
810.101.1b 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
810.101.2a1 R 2 0 2 2 2 2 2 2 2 2 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0 0 0 0 0
810.101.2a2 R 2 0 2 2 2 2 2 2 2 2 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0 0 0 0 0
810.101.5a1 C 5 1 −4−3ζ3 −1+3ζ3 −1−3ζ3 2+3ζ3 −1 −1 2+3ζ3 −1−3ζ3 −1 2 2 −1 0 0 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3
810.101.5a2 C 5 1 −1+3ζ3 −4−3ζ3 2+3ζ3 −1−3ζ3 −1 −1 −1−3ζ3 2+3ζ3 −1 2 2 −1 0 0 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1
810.101.5b1 C 5 1 −1 −1 −1−3ζ3 2+3ζ3 −1+3ζ3 −4−3ζ3 −1−3ζ3 2+3ζ3 2 −1 −1 2 0 0 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
810.101.5b2 C 5 1 −1 −1 2+3ζ3 −1−3ζ3 −4−3ζ3 −1+3ζ3 2+3ζ3 −1−3ζ3 2 −1 −1 2 0 0 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
810.101.5c1 C 5 1 −1−3ζ3 2+3ζ3 −1+3ζ3 −4−3ζ3 2+3ζ3 −1−3ζ3 −1 −1 2 2 −1 −1 0 0 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1
810.101.5c2 C 5 1 2+3ζ3 −1−3ζ3 −4−3ζ3 −1+3ζ3 −1−3ζ3 2+3ζ3 −1 −1 2 2 −1 −1 0 0 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1
810.101.5d1 C 5 1 −1−3ζ3 2+3ζ3 −1 −1 −1−3ζ3 2+3ζ3 −4−3ζ3 −1+3ζ3 −1 −1 2 2 0 0 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
810.101.5d2 C 5 1 2+3ζ3 −1−3ζ3 −1 −1 2+3ζ3 −1−3ζ3 −1+3ζ3 −4−3ζ3 −1 −1 2 2 0 0 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
810.101.5e1 C 5 −1 −4−3ζ3 −1+3ζ3 −1−3ζ3 2+3ζ3 −1 −1 2+3ζ3 −1−3ζ3 −1 2 2 −1 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3
810.101.5e2 C 5 −1 −1+3ζ3 −4−3ζ3 2+3ζ3 −1−3ζ3 −1 −1 −1−3ζ3 2+3ζ3 −1 2 2 −1 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1
810.101.5f1 C 5 −1 −1 −1 −1−3ζ3 2+3ζ3 −1+3ζ3 −4−3ζ3 −1−3ζ3 2+3ζ3 2 −1 −1 2 0 0 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
810.101.5f2 C 5 −1 −1 −1 2+3ζ3 −1−3ζ3 −4−3ζ3 −1+3ζ3 2+3ζ3 −1−3ζ3 2 −1 −1 2 0 0 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
810.101.5g1 C 5 −1 −1−3ζ3 2+3ζ3 −1+3ζ3 −4−3ζ3 2+3ζ3 −1−3ζ3 −1 −1 2 2 −1 −1 0 0 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1
810.101.5g2 C 5 −1 2+3ζ3 −1−3ζ3 −4−3ζ3 −1+3ζ3 −1−3ζ3 2+3ζ3 −1 −1 2 2 −1 −1 0 0 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1
810.101.5h1 C 5 −1 −1−3ζ3 2+3ζ3 −1 −1 −1−3ζ3 2+3ζ3 −4−3ζ3 −1+3ζ3 −1 −1 2 2 0 0 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
810.101.5h2 C 5 −1 2+3ζ3 −1−3ζ3 −1 −1 2+3ζ3 −1−3ζ3 −1+3ζ3 −4−3ζ3 −1 −1 2 2 0 0 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
810.101.10a R 10 0 −2 −2 −2 −2 4 4 4 4 1 −5 1 −2 0 0 0 0 0 0 0 0 0 0
810.101.10b R 10 0 −2 −2 4 4 4 4 −2 −2 −2 1 −5 1 0 0 0 0 0 0 0 0 0 0
810.101.10c R 10 0 4 4 −2 −2 −2 −2 4 4 −5 1 −2 1 0 0 0 0 0 0 0 0 0 0
810.101.10d R 10 0 4 4 4 4 −2 −2 −2 −2 1 −2 1 −5 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);