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Properties

Label 12T285
Order \(23040\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $285$
CHM label :  $[2^{5}]S(6)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,3)(2,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
720:  $S_6$
11520:  16T1753

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $S_6$

Low degree siblings

12T285, 20T531 x 4, 24T12618, 32T1120025 x 4, 40T14226 x 2, 40T14229, 40T14232 x 2, 40T14235, 40T14236 x 2

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 1,12)( 2, 3)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $( 8,10)( 9,11)$
$ 4, 2, 1, 1, 1, 1, 1, 1 $ $120$ $4$ $( 1,12)( 8,11, 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $180$ $2$ $( 1,12)( 2, 3)( 8,10)( 9,11)$
$ 4, 2, 2, 2, 1, 1 $ $120$ $4$ $( 1,12)( 2, 3)( 4, 5)( 8,11, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $30$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8,10)( 9,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $160$ $3$ $( 6, 8,10)( 7, 9,11)$
$ 6, 2, 1, 1, 1, 1 $ $480$ $6$ $( 1,12)( 6, 8,11, 7, 9,10)$
$ 3, 3, 2, 2, 1, 1 $ $480$ $6$ $( 1,12)( 2, 3)( 6, 8,10)( 7, 9,11)$
$ 6, 2, 2, 2 $ $160$ $6$ $( 1,12)( 2, 3)( 4, 5)( 6, 8,11, 7, 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $180$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 4, 2, 2, 2, 1, 1 $ $720$ $4$ $( 1,12)( 4, 6)( 5, 7)( 8,11, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $180$ $2$ $( 1,12)( 2, 3)( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $180$ $4$ $( 4, 6, 5, 7)( 8,11, 9,10)$
$ 4, 4, 2, 2 $ $180$ $4$ $( 1,12)( 2, 3)( 4, 6, 5, 7)( 8,11, 9,10)$
$ 4, 4, 1, 1, 1, 1 $ $720$ $4$ $( 4, 6, 8,10)( 5, 7, 9,11)$
$ 8, 2, 1, 1 $ $1440$ $8$ $( 1,12)( 4, 6, 8,11, 5, 7, 9,10)$
$ 4, 4, 2, 2 $ $720$ $4$ $( 1,12)( 2, 3)( 4, 6, 8,10)( 5, 7, 9,11)$
$ 3, 3, 2, 2, 1, 1 $ $960$ $6$ $( 2, 4)( 3, 5)( 6, 8,10)( 7, 9,11)$
$ 6, 2, 2, 2 $ $960$ $6$ $( 1,12)( 2, 4)( 3, 5)( 6, 8,11, 7, 9,10)$
$ 6, 4, 1, 1 $ $960$ $12$ $( 2, 4, 3, 5)( 6, 8,11, 7, 9,10)$
$ 4, 3, 3, 2 $ $960$ $12$ $( 1,12)( 2, 4, 3, 5)( 6, 8,10)( 7, 9,11)$
$ 5, 5, 1, 1 $ $2304$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$
$ 10, 2 $ $2304$ $10$ $( 1,12)( 2, 4, 6, 8,11, 3, 5, 7, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 4, 4, 2, 2 $ $360$ $4$ $( 1, 3,12, 2)( 4, 6)( 5, 7)( 8,11, 9,10)$
$ 4, 4, 2, 2 $ $720$ $4$ $( 1, 3)( 2,12)( 4, 6, 8,10)( 5, 7, 9,11)$
$ 4, 4, 2, 2 $ $720$ $4$ $( 1, 3)( 2,12)( 4, 6, 9,10)( 5, 7, 8,11)$
$ 8, 4 $ $1440$ $8$ $( 1, 3,12, 2)( 4, 6, 8,11, 5, 7, 9,10)$
$ 3, 3, 3, 3 $ $640$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$
$ 6, 6 $ $640$ $6$ $( 1, 3, 5,12, 2, 4)( 6, 8,11, 7, 9,10)$
$ 6, 6 $ $1920$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 6 $ $1920$ $6$ $( 1, 3, 5, 7, 8,11)( 2, 4, 6, 9,10,12)$

Group invariants

Order:  $23040=2^{9} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.