↑

Properties

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

Related objects

Downloads

Learn more

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

|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 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 $ $20$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4, 9,14)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $324$ $5$ $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 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 $ $90$ $6$ $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1, 6,11)( 2,15)( 3,14,13, 9, 8, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15,12,10, 7, 5)( 3, 9, 8,14,13, 4)$
$ 12, 2, 1 $ $135$ $12$ $( 2, 3,15,14, 7,13, 5, 9,12, 8,10, 4)( 6,11)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 1,11)( 2, 3,15, 4)( 5,14, 7,13)( 8,10, 9,12)$
$ 12, 2, 1 $ $135$ $12$ $( 1, 6)( 2, 3,15, 9,12, 8,10,14, 7,13, 5, 4)$
$ 12, 2, 1 $ $135$ $12$ $( 2, 4,15,13, 7,14, 5, 8,12, 9,10, 3)( 6,11)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 1,11)( 2, 9,10, 3)( 4,15,13, 7)( 5, 8,12,14)$
$ 12, 2, 1 $ $135$ $12$ $( 1, 6)( 2,14, 5, 8,12, 4,15,13, 7, 9,10, 3)$

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 3B 3C-1 3C1 3D 3E 2A 2A 5A 3A1 3A-1 3C-1 3B 3C1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 4A-1 4A1 5A 2A 2A 2A 2A 2A 4A1 4A-1 4A1 4A-1
5 P 1A 2A 3A-1 3A1 3B 3C-1 3C1 3D 3E 4A1 4A-1 1A 6A-1 6A1 6C1 6B 6C-1 12A5 12A-5 12A1 12A-1
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);