Properties

Label 15T24
Order \(360\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5 \times C_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $24$
Group :  $S_5 \times C_3$
CHM label :  $S(5)[x]3$
Parity:  $-1$
Primitive:  No
Generators:  (1,4)(6,9)(11,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $S_5$

Low degree siblings

18T144, 30T90, 30T98, 30T103, 36T550, 45T44

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 5,14)( 9,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 3, 9,15)( 4,10,13)( 5, 8,14)$
$ 4, 4, 4, 1, 1, 1 $ $30$ $4$ $( 2, 5, 8,14)( 3, 9,12,15)( 4, 7,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 6, 3, 3, 3 $ $20$ $6$ $( 1, 2, 3)( 4, 5, 9,10,14,15)( 6, 7, 8)(11,12,13)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 2, 3)( 4,14, 9)( 5,15,10)( 6, 7, 8)(11,12,13)$
$ 15 $ $24$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
$ 12, 3 $ $30$ $12$ $( 1, 2, 3, 4,11,12,13,14, 6, 7, 8, 9)( 5,15,10)$
$ 6, 6, 3 $ $15$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$
$ 6, 3, 3, 3 $ $10$ $6$ $( 1, 2, 6, 7,11,12)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 3, 2)( 4, 9,14)( 5,10,15)( 6, 8, 7)(11,13,12)$
$ 6, 3, 3, 3 $ $20$ $6$ $( 1, 3, 2)( 4,15,14,10, 9, 5)( 6, 8, 7)(11,13,12)$
$ 12, 3 $ $30$ $12$ $( 1, 3, 5, 7, 6, 8,10,12,11,13,15, 2)( 4, 9,14)$
$ 15 $ $24$ $15$ $( 1, 3, 5, 4,12,11,13,15,14, 7, 6, 8,10, 9, 2)$
$ 6, 6, 3 $ $15$ $6$ $( 1, 3,11,13, 6, 8)( 2, 4,12,14, 7, 9)( 5,10,15)$
$ 6, 3, 3, 3 $ $10$ $6$ $( 1, 3,11,13, 6, 8)( 2, 7,12)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $24$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 3, 3, 3, 2, 2, 2 $ $20$ $6$ $( 1, 4,13)( 2, 5)( 3, 6, 9)( 7,10)( 8,11,14)(12,15)$
$ 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 119]
Character table: Data not available.