↑

Properties

Label 12T258
Order \(3888\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $258$
CHM label :  $[3^{4}:2]S(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(6,9), (1,5)(2,10)(4,8)(7,11), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
24:  $S_4$
36:  $S_3^2$
48:  $S_4\times C_2$
144:  12T83
1296:  $S_3\wr S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: $S_4$

Degree 6: None

Low degree siblings

12T258, 18T448, 24T7264 x 2, 24T7265 x 2, 24T7266 x 2, 24T7271 x 2, 27T532, 36T4795, 36T4796, 36T4797 x 2, 36T4798 x 2, 36T4799, 36T4817, 36T4821 x 2, 36T4952

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3 $ $6$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 1, 5, 9)( 2, 6,10)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 6, 6 $ $54$ $6$ $( 1,12, 9, 8, 5, 4)( 2,11, 6, 3,10, 7)$
$ 3, 3, 3, 1, 1, 1 $ $72$ $3$ $( 2,11, 8)( 3,12, 6)( 4,10, 7)$
$ 9, 3 $ $144$ $9$ $( 1, 5, 9)( 2, 7,12,10, 3, 8, 6,11, 4)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3,12)( 4, 7)( 8,11)$
$ 6, 3, 3 $ $36$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 8, 7,12,11, 4)$
$ 6, 3, 1, 1, 1 $ $72$ $6$ $( 2, 6,10)( 3, 8,11, 4, 7,12)$
$ 3, 3, 2, 2, 2 $ $36$ $6$ $( 1, 5, 9)( 2,10, 6)( 3, 4)( 7, 8)(11,12)$
$ 4, 4, 4 $ $162$ $4$ $( 1, 4,10, 7)( 2,11, 5, 8)( 3, 9,12, 6)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 4,12, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $24$ $3$ $( 1, 9, 5)( 2,10, 6)( 3, 7,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 2, 2, 2 $ $108$ $6$ $( 1, 4, 9,12, 5, 8)( 2,11)( 3, 6)( 7,10)$
$ 6, 6 $ $54$ $6$ $( 1,12, 5, 4, 9, 8)( 2,11, 6, 3,10, 7)$
$ 9, 1, 1, 1 $ $144$ $9$ $( 2,11, 8,10, 7, 4, 6, 3,12)$
$ 9, 3 $ $144$ $9$ $( 1, 5, 9)( 2, 7,12, 6,11, 4,10, 3, 8)$
$ 3, 3, 3, 3 $ $144$ $3$ $( 1, 9, 5)( 2, 3, 4)( 6, 7, 8)(10,11,12)$
$ 6, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 3,12,11, 8, 7, 4)$
$ 3, 3, 2, 2, 2 $ $36$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 8)( 4,11)( 7,12)$
$ 6, 3, 3 $ $36$ $6$ $( 1, 9, 5)( 2,10, 6)( 3, 4, 7, 8,11,12)$
$ 6, 3, 1, 1, 1 $ $72$ $6$ $( 2, 6,10)( 3, 8, 7,12,11, 4)$
$ 6, 3, 3 $ $72$ $6$ $( 1, 5, 9)( 2,10, 6)( 3, 4,11,12, 7, 8)$
$ 3, 2, 2, 2, 1, 1, 1 $ $72$ $6$ $( 1, 9, 5)( 3,12)( 4, 7)( 8,11)$
$ 12 $ $324$ $12$ $( 1, 4, 6, 3, 9,12, 2,11, 5, 8,10, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$ 6, 2, 2, 2 $ $108$ $6$ $( 1, 4)( 2,11,10, 3, 6, 7)( 5,12)( 8, 9)$
$ 6, 6 $ $108$ $6$ $( 1,12, 9, 4, 5, 8)( 2, 7, 6, 3,10,11)$
$ 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 8)( 2,11)( 3,10)( 4, 5)( 6, 7)( 9,12)$
$ 6, 3, 2, 1 $ $648$ $6$ $( 2,11, 4,10, 3,12)( 5, 9)( 6, 7, 8)$
$ 6, 2, 2, 1, 1 $ $324$ $6$ $( 3,12,11, 4, 7, 8)( 5, 9)( 6,10)$
$ 2, 2, 2, 2, 2, 1, 1 $ $162$ $2$ $( 2, 6)( 3, 8)( 4, 7)( 5, 9)(11,12)$
$ 12 $ $324$ $12$ $( 1, 4,10, 3, 9, 8, 6, 7, 5,12, 2,11)$
$ 4, 4, 4 $ $162$ $4$ $( 1,12, 6, 3)( 2, 7, 9, 4)( 5, 8,10,11)$

Group invariants

Order:  $3888=2^{4} \cdot 3^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.