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Properties

Label 15T29
Degree $15$
Order $720$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5 \times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 29);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_5 \times S_3$
CHM label:   $S(5)[x]S(3)$
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:  (1,11)(2,7)(4,14)(5,10)(8,13), (1,4)(6,9)(11,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$120$:  $S_5$
$240$:  $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $S_5$

Low degree siblings

18T227, 30T165, 30T167, 30T170, 30T174, 30T178, 36T1249, 36T1250, 36T1251, 45T94

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, 3, 3, 3 $ $2$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 4)( 6, 9)(11,14)$
$ 6, 3, 3, 3 $ $20$ $6$ $( 1, 9,11, 4, 6,14)( 2, 7,12)( 3, 8,13)( 5,10,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $30$ $2$ $( 1, 4)( 2,12)( 3, 8)( 5,15)( 6,14)( 9,11)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 1, 4)( 2, 5)( 6, 9)( 7,10)(11,14)(12,15)$
$ 6, 6, 3 $ $30$ $6$ $( 1, 9,11, 4, 6,14)( 2,10,12, 5, 7,15)( 3, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 1, 4)( 2,15)( 3, 8)( 5,12)( 6,14)( 7,10)( 9,11)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 4, 7)( 2,11,14)( 6, 9,12)$
$ 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 9, 2)( 3, 8,13)( 4,12,11)( 5,10,15)( 6,14, 7)$
$ 6, 3, 2, 2, 1, 1 $ $60$ $6$ $( 1, 4, 7)( 2, 6,14,12,11, 9)( 3, 8)( 5,15)$
$ 3, 3, 3, 2, 2, 2 $ $20$ $6$ $( 1, 4, 7)( 2,11,14)( 3,15)( 5, 8)( 6, 9,12)(10,13)$
$ 6, 3, 3, 3 $ $40$ $6$ $( 1, 9, 2)( 3, 5,13,15, 8,10)( 4,12,11)( 6,14, 7)$
$ 6, 3, 2, 2, 2 $ $60$ $6$ $( 1, 4, 7)( 2, 6,14,12,11, 9)( 3, 5)( 8,15)(10,13)$
$ 4, 4, 4, 1, 1, 1 $ $30$ $4$ $( 1, 4, 7,10)( 2, 5,11,14)( 6, 9,12,15)$
$ 12, 3 $ $60$ $12$ $( 1, 9, 2,10, 6,14, 7,15,11, 4,12, 5)( 3, 8,13)$
$ 4, 4, 4, 2, 1 $ $90$ $4$ $( 1, 4, 7,10)( 2,15,11, 9)( 3, 8)( 5, 6,14,12)$
$ 5, 5, 5 $ $24$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $48$ $15$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$
$ 10, 5 $ $72$ $10$ $( 1, 4, 7,10,13)( 2,15, 8, 6,14,12, 5, 3,11, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  720.767
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 5A 6A 6B 6C 6D 6E 6F 10A 12A 15A
Size 1 3 10 15 30 45 2 20 40 30 90 24 20 20 30 40 60 60 72 60 48
2 P 1A 1A 1A 1A 1A 1A 3A 3B 3C 2C 2C 5A 3B 3A 3A 3C 3B 3B 5A 6C 15A
3 P 1A 2A 2B 2C 2D 2E 1A 1A 1A 4A 4B 5A 2B 2B 2C 2B 2A 2D 10A 4A 5A
5 P 1A 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 1A 6A 6B 6C 6D 6E 6F 2A 12A 3A
Type
720.767.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
720.767.1b R 1 −1 −1 1 1 −1 1 1 1 −1 1 1 −1 −1 1 −1 −1 1 −1 −1 1
720.767.1c R 1 −1 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 1 1
720.767.1d R 1 1 −1 1 −1 1 1 1 1 −1 −1 1 −1 −1 1 −1 1 −1 1 −1 1
720.767.2a R 2 0 2 2 0 0 −1 2 −1 2 0 2 2 −1 −1 −1 0 0 0 −1 −1
720.767.2b R 2 0 −2 2 0 0 −1 2 −1 −2 0 2 −2 1 −1 1 0 0 0 1 −1
720.767.4a R 4 4 2 0 2 0 4 1 1 0 0 −1 −1 2 0 −1 1 −1 −1 0 −1
720.767.4b R 4 4 −2 0 −2 0 4 1 1 0 0 −1 1 −2 0 1 1 1 −1 0 −1
720.767.4c R 4 −4 −2 0 2 0 4 1 1 0 0 −1 1 −2 0 1 −1 −1 1 0 −1
720.767.4d R 4 −4 2 0 −2 0 4 1 1 0 0 −1 −1 2 0 −1 −1 1 1 0 −1
720.767.5a R 5 5 −1 1 −1 1 5 −1 −1 1 1 0 −1 −1 1 −1 −1 −1 0 1 0
720.767.5b R 5 5 1 1 1 1 5 −1 −1 −1 −1 0 1 1 1 1 −1 1 0 −1 0
720.767.5c R 5 −5 −1 1 1 −1 5 −1 −1 1 −1 0 −1 −1 1 −1 1 1 0 1 0
720.767.5d R 5 −5 1 1 −1 −1 5 −1 −1 −1 1 0 1 1 1 1 1 −1 0 −1 0
720.767.6a R 6 6 0 −2 0 −2 6 0 0 0 0 1 0 0 −2 0 0 0 1 0 1
720.767.6b R 6 −6 0 −2 0 2 6 0 0 0 0 1 0 0 −2 0 0 0 −1 0 1
720.767.8a R 8 0 4 0 0 0 −4 2 −1 0 0 −2 −2 −2 0 1 0 0 0 0 1
720.767.8b R 8 0 −4 0 0 0 −4 2 −1 0 0 −2 2 2 0 −1 0 0 0 0 1
720.767.10a R 10 0 −2 2 0 0 −5 −2 1 2 0 0 −2 1 −1 1 0 0 0 −1 0
720.767.10b R 10 0 2 2 0 0 −5 −2 1 −2 0 0 2 −1 −1 −1 0 0 0 1 0
720.767.12a R 12 0 0 −4 0 0 −6 0 0 0 0 2 0 0 2 0 0 0 0 0 −1

magma: CharacterTable(G);