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Properties

Label 15T51
Order \(3000\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $51$
CHM label :  $[1/2.D(5)^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,7)(4,14)(5,10)(8,13), (1,4)(2,8)(7,13)(11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

20T268, 30T414, 30T416, 30T429, 30T430, 30T439, 40T2383

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 $ $1$ $1$ $()$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 6, 9,12,15)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $24$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $75$ $2$ $( 5,14)( 6,15)( 8,11)( 9,12)$
$ 5, 2, 2, 2, 2, 1, 1 $ $150$ $10$ $( 1, 4, 7,10,13)( 5,14)( 6,15)( 8,11)( 9,12)$
$ 5, 2, 2, 2, 2, 1, 1 $ $150$ $10$ $( 1, 7,13, 4,10)( 5,14)( 6,15)( 8,11)( 9,12)$
$ 3, 3, 3, 3, 3 $ $200$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $200$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 15 $ $200$ $15$ $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$
$ 15 $ $200$ $15$ $( 1,11, 3,13, 8,15,10, 5,12, 7, 2, 9, 4,14, 6)$
$ 15 $ $200$ $15$ $( 1,11,15,10, 5, 9, 4,14, 3,13, 8,12, 7, 2, 6)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $30$ $2$ $( 1,11)( 2, 7)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3, 6, 9,12,15)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3, 9,15, 6,12)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3,15,12, 9, 6)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3,12, 6,15, 9)( 4,14)( 5,10)( 8,13)$
$ 10, 1, 1, 1, 1, 1 $ $60$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)$
$ 10, 5 $ $60$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 6, 9,12,15)$
$ 10, 5 $ $60$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 9,15, 6,12)$
$ 10, 5 $ $60$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3,15,12, 9, 6)$
$ 10, 5 $ $60$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3,12, 6,15, 9)$
$ 10, 1, 1, 1, 1, 1 $ $60$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)$
$ 10, 5 $ $60$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3, 6, 9,12,15)$
$ 10, 5 $ $60$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3, 9,15, 6,12)$
$ 10, 5 $ $60$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,15,12, 9, 6)$
$ 10, 5 $ $60$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,12, 6,15, 9)$
$ 4, 4, 2, 2, 2, 1 $ $750$ $4$ $( 1, 8,13,11)( 2, 7)( 4, 5,10,14)( 6,15)( 9,12)$

Group invariants

Order:  $3000=2^{3} \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.