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Properties

Label 15T42
Degree $15$
Order $1620$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(15, 42);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:F_5$
CHM label:   $1/2[3^{4}:2]F(5)$
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:  (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $F_5$

Low degree siblings

15T41, 30T295, 30T296, 30T298, 30T299, 30T302, 45T204, 45T210

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{15}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{3}$ $45$ $2$ $6$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
3A1 $3^{4},1^{3}$ $5$ $3$ $8$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)$
3A-1 $3^{4},1^{3}$ $5$ $3$ $8$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$
3B $3^{3},1^{6}$ $10$ $3$ $6$ $( 1, 6,11)( 3, 8,13)( 5,10,15)$
3C1 $3^{5}$ $10$ $3$ $10$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$
3C-1 $3^{3},1^{6}$ $10$ $3$ $6$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$
3D $3^{2},1^{9}$ $20$ $3$ $4$ $( 1,11, 6)( 4, 9,14)$
3E $3^{4},1^{3}$ $20$ $3$ $8$ $( 1,11, 6)( 3, 8,13)( 4,14, 9)( 5,10,15)$
4A1 $4^{3},2,1$ $135$ $4$ $10$ $( 1,11)( 2, 4, 5,13)( 3,12, 9,15)( 7,14,10, 8)$
4A-1 $4^{3},2,1$ $135$ $4$ $10$ $( 1,11)( 2,13, 5, 4)( 3,15, 9,12)( 7, 8,10,14)$
5A $5^{3}$ $324$ $5$ $12$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
6A1 $6^{2},1^{3}$ $45$ $6$ $10$ $( 1, 4, 6, 9,11,14)( 2,13,12, 8, 7, 3)$
6A-1 $6^{2},1^{3}$ $45$ $6$ $10$ $( 1,14,11, 9, 6, 4)( 2, 3, 7, 8,12,13)$
6B $6^{2},3$ $90$ $6$ $12$ $( 1, 4,11,14, 6, 9)( 2, 3,12,13, 7, 8)( 5,15,10)$
6C1 $6,3,2^{3}$ $90$ $6$ $10$ $( 1, 5,11,15, 6,10)( 2,14)( 3, 8,13)( 4, 7)( 9,12)$
6C-1 $6,3,2^{3}$ $90$ $6$ $10$ $( 1,10, 6,15,11, 5)( 2,14)( 3,13, 8)( 4, 7)( 9,12)$
12A1 $12,2,1$ $135$ $12$ $12$ $( 1, 3, 4, 2, 6,13, 9,12,11, 8,14, 7)( 5,15)$
12A-1 $12,2,1$ $135$ $12$ $12$ $( 1,12, 4, 8, 6, 7, 9, 3,11, 2,14,13)( 5,15)$
12A5 $12,2,1$ $135$ $12$ $12$ $( 1, 7,14, 8,11,12, 9,13, 6, 2, 4, 3)( 5,15)$
12A-5 $12,2,1$ $135$ $12$ $12$ $( 1,13,14, 2,11, 3, 9, 7, 6, 8, 4,12)( 5,15)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1620=2^{2} \cdot 3^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1620.421
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 3D 3E 4A1 4A-1 5A 6A1 6A-1 6B 6C1 6C-1 12A1 12A-1 12A5 12A-5
Size 1 45 5 5 10 10 10 20 20 135 135 324 45 45 90 90 90 135 135 135 135
2 P 1A 1A 3A-1 3A1 3C1 3B 3C-1 3D 3E 2A 2A 5A 3A1 3A-1 3B 3C1 3C-1 6A1 6A1 6A-1 6A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 4A-1 4A1 5A 2A 2A 2A 2A 2A 4A-1 4A1 4A1 4A-1
5 P 1A 2A 3A-1 3A1 3C1 3B 3C-1 3D 3E 4A1 4A-1 1A 6A-1 6A1 6B 6C-1 6C1 12A-1 12A5 12A1 12A-5
Type
1620.421.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1620.421.1b R 1 1 1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
1620.421.1c1 C 1 −1 1 1 1 1 1 1 1 −i i 1 −1 −1 −1 −1 −1 i −i i −i
1620.421.1c2 C 1 −1 1 1 1 1 1 1 1 i −i 1 −1 −1 −1 −1 −1 −i i −i i
1620.421.4a R 4 0 4 4 4 4 4 4 4 0 0 −1 0 0 0 0 0 0 0 0 0
1620.421.5a1 C 5 1 −4−3ζ3 −1+3ζ3 −1 2+3ζ3 −1−3ζ3 2 −1 1 1 0 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
1620.421.5a2 C 5 1 −1+3ζ3 −4−3ζ3 −1 −1−3ζ3 2+3ζ3 2 −1 1 1 0 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
1620.421.5b1 C 5 1 −4−3ζ3 −1+3ζ3 −1 2+3ζ3 −1−3ζ3 2 −1 −1 −1 0 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
1620.421.5b2 C 5 1 −1+3ζ3 −4−3ζ3 −1 −1−3ζ3 2+3ζ3 2 −1 −1 −1 0 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
1620.421.5c1 C 5 −1 −1−3ζ122 −4+3ζ122 −1 −1+3ζ122 2−3ζ122 2 −1 −ζ123 ζ123 0 ζ122 −ζ124 −1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12
1620.421.5c2 C 5 −1 −4+3ζ122 −1−3ζ122 −1 2−3ζ122 −1+3ζ122 2 −1 ζ123 −ζ123 0 −ζ124 ζ122 −1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125
1620.421.5c3 C 5 −1 −1−3ζ122 −4+3ζ122 −1 −1+3ζ122 2−3ζ122 2 −1 ζ123 −ζ123 0 ζ122 −ζ124 −1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12
1620.421.5c4 C 5 −1 −4+3ζ122 −1−3ζ122 −1 2−3ζ122 −1+3ζ122 2 −1 −ζ123 ζ123 0 −ζ124 ζ122 −1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125
1620.421.10a R 10 2 −2 −2 −5 1 1 −2 4 0 0 0 2 2 −1 −1 −1 0 0 0 0
1620.421.10b S 10 −2 −2 −2 −5 1 1 −2 4 0 0 0 −2 −2 1 1 1 0 0 0 0
1620.421.10c1 C 10 2 −2−6ζ3 4+6ζ3 1 −5−3ζ3 −2+3ζ3 1 1 0 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 0 0
1620.421.10c2 C 10 2 4+6ζ3 −2−6ζ3 1 −2+3ζ3 −5−3ζ3 1 1 0 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 0 0
1620.421.10d1 C 10 −2 −2−6ζ3 4+6ζ3 1 −5−3ζ3 −2+3ζ3 1 1 0 0 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0 0 0
1620.421.10d2 C 10 −2 4+6ζ3 −2−6ζ3 1 −2+3ζ3 −5−3ζ3 1 1 0 0 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0 0 0
1620.421.20a R 20 0 8 8 −4 2 2 −1 −4 0 0 0 0 0 0 0 0 0 0 0 0
1620.421.20b R 20 0 −4 −4 8 2 2 −4 −1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);