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Properties

Label 15T23
Degree $15$
Order $360$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5 \times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(15, 23);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_5 \times S_3$
CHM label:   $A(5)[x]S(3)$
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), (1,13)(2,14)(3,6)(4,7)(8,11)(9,12), (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)
$2$:  $C_2$
$6$:  $S_3$
$60$:  $A_5$
$120$:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $A_5$

Low degree siblings

18T145, 30T85, 30T94, 30T102, 36T551, 36T552, 36T553, 45T40

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, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 3, 9,15)( 4,10,13)( 5, 8,14)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 2, 3)( 4,10)( 5, 9)( 6,11)( 7,13)( 8,12)(14,15)$
$ 6, 3, 2, 2, 1, 1 $ $60$ $6$ $( 2, 3, 5,12, 8,15)( 6,11)( 7,13,10)( 9,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
$ 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2, 3)( 4,14, 9)( 5,15,10)( 6, 7, 8)(11,12,13)$
$ 15 $ $24$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
$ 15 $ $24$ $15$ $( 1, 2, 3,10,14, 6, 7, 8,15, 4,11,12,13, 5, 9)$
$ 10, 5 $ $36$ $10$ $( 1, 2, 4, 5,13,11, 7,14,10, 8)( 3, 6,12, 9,15)$
$ 10, 5 $ $36$ $10$ $( 1, 2, 4, 8,10,11, 7,14,13, 5)( 3,15, 6,12, 9)$
$ 6, 6, 3 $ $30$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4,10, 7,13)( 2, 8,11,14, 5)( 3, 6, 9,15,12)$
$ 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  360.121
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 5A1 5A2 6A 6B 10A1 10A3 15A1 15A2
Size 1 3 15 45 2 20 40 12 12 30 60 36 36 24 24
2 P 1A 1A 1A 1A 3A 3B 3C 5A2 5A1 3A 3B 5A1 5A2 15A2 15A1
3 P 1A 2A 2B 2C 1A 1A 1A 5A2 5A1 2B 2A 10A3 10A1 5A1 5A2
5 P 1A 2A 2B 2C 3A 3B 3C 1A 1A 6A 6B 2A 2A 3A 3A
Type
360.121.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
360.121.1b R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 1 1
360.121.2a R 2 0 2 0 −1 2 −1 2 2 −1 0 0 0 −1 −1
360.121.3a1 R 3 3 −1 −1 3 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
360.121.3a2 R 3 3 −1 −1 3 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
360.121.3b1 R 3 −3 −1 1 3 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
360.121.3b2 R 3 −3 −1 1 3 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
360.121.4a R 4 4 0 0 4 1 1 −1 −1 0 1 −1 −1 −1 −1
360.121.4b R 4 −4 0 0 4 1 1 −1 −1 0 −1 1 1 −1 −1
360.121.5a R 5 5 1 1 5 −1 −1 0 0 1 −1 0 0 0 0
360.121.5b R 5 −5 1 −1 5 −1 −1 0 0 1 1 0 0 0 0
360.121.6a1 R 6 0 −2 0 −3 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 1 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
360.121.6a2 R 6 0 −2 0 −3 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 1 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
360.121.8a R 8 0 0 0 −4 2 −1 −2 −2 0 0 0 0 1 1
360.121.10a R 10 0 2 0 −5 −2 1 0 0 −1 0 0 0 0 0

magma: CharacterTable(G);