↑

Properties

Label 12T193
Order \(768\)
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 $t$ :  $193$
CHM label :  $[2^{6}]D(6)=2wrD(6)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,11)(2,8)(3,9)(4,6)(5,7)(10,12)
$|\Aut(F/K)|$:  $2$

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: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Low degree siblings

12T186 x 4, 12T193 x 3, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 2, 3)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 8, 9)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)( 2, 3)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 2, 3)( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 1,12)( 4, 5)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 1,12)( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 3, 3 $ $64$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,10)( 3, 7,11)$
$ 6, 6 $ $32$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,11, 3, 7,10)$
$ 6, 6 $ $64$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 12 $ $64$ $12$ $( 1, 3, 5, 7, 9,11,12, 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,11, 5,10)$
$ 4, 4, 4 $ $8$ $4$ $( 1, 7,12, 6)( 2, 8, 3, 9)( 4,11, 5,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 2,11, 3,10)( 4, 8)( 5, 9)$
$ 4, 2, 2, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2,11, 3,10)( 4, 8)( 5, 9)$
$ 4, 2, 2, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2,10)( 3,11)( 4, 9, 5, 8)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2,11, 3,10)( 4, 9, 5, 8)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2,11, 3,10)( 4, 9, 5, 8)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)( 6, 7)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,12)( 2,11, 3,10)( 4, 8)( 5, 9)( 6, 7)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1,12)( 2,11, 3,10)( 4, 9, 5, 8)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $48$ $4$ $( 1, 3,12, 2)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 3)( 2,12)( 4,11, 5,10)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $48$ $4$ $( 1, 3,12, 2)( 4,11, 5,10)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 3,12, 2)( 4,10)( 5,11)( 6, 9, 7, 8)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 3,12, 2)( 4,11, 5,10)( 6, 9, 7, 8)$

Group invariants

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