Properties

Label 15T29
Order \(720\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5 \times S_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $29$
Group :  $S_5 \times S_3$
CHM label :  $S(5)[x]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

Cycle 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 $ $72$ $10$ $( 1, 4, 7,10,13)( 2,15, 8, 6,14,12, 5, 3,11, 9)$

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [720, 767]
Character table: Data not available.