# Properties

 Label 12T193 Degree $12$ Order $768$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

# Related objects

## 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) $\card{\Aut(F/K)}$: $2$ 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)

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

## Group invariants

 Order: $768=2^{8} \cdot 3$ Cyclic: no Abelian: no Solvable: yes Label: 768.1087581
 Character table: not available.