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Properties

Label 12T186
Order \(768\)
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 $t$ :  $186$
CHM label :  $[1/4.eD(4)^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,12)(6,9), (3,9)(6,12), (1,7)(3,9)(5,11), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 3
16:  $D_4\times C_2$
24:  $S_4$, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 3, 12T28
96:  12T48
192:  $V_4^2:(S_3\times C_2)$, 12T86
384:  12T136

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

12T186 x 3, 12T193 x 4, 16T1053 x 4, 24T1557, 24T1598, 24T1767, 24T1844, 24T1882, 24T1911, 24T2208 x 2, 24T2209 x 2, 24T2487 x 2, 24T2488 x 2, 24T2552 x 2, 24T2553 x 4, 24T2554 x 2, 24T2555 x 2, 24T2556 x 2, 24T2557 x 2, 24T2558 x 2, 24T2595 x 2, 24T2596 x 2, 24T2597 x 4, 24T2598 x 4, 24T2599 x 4, 24T2600 x 4, 24T2601 x 4, 24T2602 x 4, 24T2603 x 4, 24T2604 x 4, 24T2605 x 2, 24T2606 x 2, 24T2607 x 2, 24T2608 x 2, 32T34686 x 2, 32T34687 x 2, 32T34688 x 2, 32T34803, 32T35039

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 $ $6$ $2$ $( 3,12)( 6, 9)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1, 4)( 3,12)( 6, 9)( 7,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1, 4)( 3, 6)( 7,10)( 9,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 1, 4)( 3, 9)( 6,12)( 7,10)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2,11)( 3,12)( 5, 8)( 6, 9)( 7,10)$
$ 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2,11)( 3, 9)( 5, 8)( 6,12)( 7,10)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 8)( 3, 9)( 5,11)( 6,12)( 7,10)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 4,10)( 5,11)( 6,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3,12, 9, 6)( 4,10)( 5,11)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1, 4, 7,10)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 4 $ $8$ $4$ $( 1, 4, 7,10)( 2,11, 8, 5)( 3,12, 9, 6)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 6 $ $64$ $6$ $( 1, 6, 2,10, 9, 5)( 3,11, 7,12, 8, 4)$
$ 6, 6 $ $32$ $6$ $( 1, 3,11, 7, 9, 5)( 2,10,12, 8, 4, 6)$
$ 6, 3, 3 $ $64$ $6$ $( 1, 9,11, 7, 3, 5)( 2, 4, 6)( 8,10,12)$
$ 12 $ $64$ $12$ $( 1, 6, 2, 4, 9,11, 7,12, 8,10, 3, 5)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 1, 5)( 2,10)( 4, 8)( 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 5)( 2,10)( 3,12)( 4, 8)( 6, 9)( 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 5)( 2,10)( 3, 9)( 4, 8)( 6,12)( 7,11)$
$ 4, 4, 1, 1, 1, 1 $ $24$ $4$ $( 1, 5, 4, 8)( 2, 7,11,10)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 5, 4, 8)( 2, 7,11,10)( 3,12)( 6, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 5, 4, 8)( 2, 7,11,10)( 3, 6)( 9,12)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 5, 4, 8)( 2, 7,11,10)( 3, 9)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 1, 2, 7, 8)( 4,11,10, 5)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2, 7, 8)( 3,12)( 4,11,10, 5)( 6, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 9)( 4,11,10, 5)( 6,12)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1,11, 7, 5)( 2, 4, 8,10)( 6,12)$
$ 4, 4, 4 $ $24$ $4$ $( 1,11, 7, 5)( 2, 4, 8,10)( 3,12, 9, 6)$
$ 4, 4, 2, 1, 1 $ $48$ $4$ $( 1,11,10, 2)( 4, 8, 7, 5)( 6,12)$
$ 4, 4, 4 $ $48$ $4$ $( 1,11,10, 2)( 3,12, 9, 6)( 4, 8, 7, 5)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 1, 2)( 4, 5)( 6,12)( 7, 8)(10,11)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12, 9, 6)( 4, 5)( 7, 8)(10,11)$

Group invariants

Order:  $768=2^{8} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [768, 1087581]
Character table: Data not available.