Properties

Label 12T42
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:D_4$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $42$
Group :  $C_3\times C_3:D_4$
CHM label :  $[3^{2}]D(4)=6wr2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(3,9)(5,11), (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $D_{4}$
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
36:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $S_3\times C_3$

Low degree siblings

12T42, 24T77, 36T19, 36T26

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 $ $1$ $1$ $()$
$ 6, 1, 1, 1, 1, 1, 1 $ $2$ $6$ $( 2, 4, 6, 8,10,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2,10, 6)( 4,12, 8)$
$ 6, 1, 1, 1, 1, 1, 1 $ $2$ $6$ $( 2,12,10, 8, 6, 4)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 12 $ $6$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 12 $ $6$ $12$ $( 1, 2,11,12, 9,10, 7, 8, 5, 6, 3, 4)$
$ 6, 6 $ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 6,10)( 4, 8,12)$
$ 6, 2, 2, 2 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 2, 2, 2 $ $2$ $6$ $( 1, 5, 9)( 2, 8)( 3, 7,11)( 4,10)( 6,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 5, 9)( 2,12,10, 8, 6, 4)( 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 3, 3, 2, 2, 2 $ $2$ $6$ $( 1, 7)( 2,10, 6)( 3, 9)( 4,12, 8)( 5,11)$
$ 6, 2, 2, 2 $ $2$ $6$ $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 3, 3 $ $2$ $6$ $( 1, 9, 5)( 2,12,10, 8, 6, 4)( 3,11, 7)$
$ 6, 6 $ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 30]
Character table: Data not available.