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Properties

Label 12T150
Order \(384\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times C_4^2:C_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $150$
Group :  $C_4\times C_4^2:C_3:C_2$
CHM label :  $[4^{3}]S(3)=4wrS(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6,9,12), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_4$, $S_3 \times C_4$
48:  $S_4\times C_2$
96:  12T53, 12T62
192:  12T95

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T150 x 3, 24T835, 24T857, 24T861, 24T1256 x 2, 24T1257 x 2, 24T1258 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 $ $1$ $1$ $()$
$ 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $4$ $( 3, 6, 9,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $4$ $( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3, 5, 6, 8, 9,11,12)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3, 8, 9)( 5, 6,11,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3,11,12, 8, 9, 5, 6)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3,12, 9, 6)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 9)( 4, 5)( 6,12)( 7, 8)(10,11)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3,12, 9, 6)( 4, 5)( 7, 8)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 12 $ $32$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3, 7, 8, 9)( 4, 5, 6,10,11,12)$
$ 12 $ $32$ $12$ $( 1, 2, 3,10,11,12, 7, 8, 9, 4, 5, 6)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 6, 9,12)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 9)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3,12, 9, 6)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 9)( 4, 5,10,11)( 6,12)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 6, 9,12)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 9)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3,12, 9, 6)$
$ 4, 4, 4 $ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3,12, 9, 6)$
$ 4, 2, 2, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 4, 7,10)( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2,11, 8, 5)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 2, 2, 2, 2 $ $3$ $4$ $( 1, 7)( 2, 8)( 3,12, 9, 6)( 4,10)( 5,11)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$
$ 4, 4, 4 $ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 5557]
Character table: Data not available.