Properties

Label 12T159
Order \(576\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $159$
CHM label :  $[2^{5}]F_{18}(6)_{2}$
Parity:  $-1$
Primitive:  No
Generators:  (1,12)(2,3), (1,5,9)(4,8,12), (2,6,10)(3,7,11), (1,2,12,3)(4,10,5,11)(6,8,7,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$
36:  $C_3\times (C_3 : C_4)$
288:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3\times C_3$

Low degree siblings

16T1027, 24T1487, 24T1488, 36T719, 36T724, 36T946, 36T947

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 $ $9$ $2$ $( 1,12)( 2, 3)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 5, 9)( 4, 8,12)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 4, 8)( 2, 3)( 5, 9,12)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 9, 5)( 4,12, 8)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 9, 4)( 2, 3)( 5,12, 8)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 8, 5,12, 9, 4)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 9)( 2,11, 6)( 3,10, 7)( 4, 8,12)$
$ 6, 6 $ $16$ $6$ $( 1, 5, 9,12, 4, 8)( 2,11, 6, 3,10, 7)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 9, 5)( 2,11, 6)( 3,10, 7)( 4,12, 8)$
$ 6, 6 $ $32$ $6$ $( 1, 9, 5,12, 8, 4)( 2,11, 6, 3,10, 7)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 9, 5)( 2, 6,11)( 3, 7,10)( 4,12, 8)$
$ 6, 6 $ $16$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 6,11, 3, 7,10)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 3,12, 2)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 12 $ $48$ $12$ $( 1, 2, 4,10, 9, 6,12, 3, 5,11, 8, 7)$
$ 12 $ $48$ $12$ $( 1, 3, 5,11, 8, 7,12, 2, 4,10, 9, 6)$
$ 12 $ $48$ $12$ $( 1, 2, 8, 7, 5,11,12, 3, 9, 6, 4,10)$
$ 12 $ $48$ $12$ $( 1, 3, 9, 6, 4,10,12, 2, 8, 7, 5,11)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8278]
Character table: Data not available.