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Properties

Label 12T188
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 $t$ :  $188$
CHM label :  $[2^{6}]A_{4}=2wrA_{4}(6)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (2,8)(3,9)(4,10)(5,11), (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:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$ x 5
24:  $A_4\times C_2$ x 5
48:  $C_2^4:C_3$
96:  $((C_2 \times D_4): C_2):C_3$ x 2, 12T56
192:  16T424
384:  16T731

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T187 x 8, 12T188 x 7, 24T2559 x 8, 24T2560 x 4, 24T2561 x 8, 24T2562 x 8, 24T2563 x 4, 24T2564 x 8, 24T2565 x 8, 24T2566 x 8, 24T2567 x 8, 24T2568 x 4, 24T2569 x 4, 24T2570 x 8, 24T2571 x 8, 24T2572 x 8, 24T2573 x 8, 24T2574 x 4, 24T2575 x 4, 24T2576 x 4, 24T2577 x 4, 32T34825 x 8

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

Group invariants

Order:  $768=2^{8} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [768, 1084555]
Character table: Data not available.