Properties

Label 12T27
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4:C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $27$
Group :  $A_4:C_4$
CHM label :  $[2]S_{4}(6)_{2}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12), (1,4,7,10)(2,3,11,12)(5,6,8,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T30, 16T62, 24T51, 24T57

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 30]
Character table:   
      2  4  3  4  3  1  3  1  3  4  4
      3  1  .  .  .  1  .  1  .  .  1

        1a 4a 2a 4b 6a 4c 3a 4d 2b 2c
     2P 1a 2a 1a 2a 3a 2c 3a 2c 1a 1a
     3P 1a 4b 2a 4a 2c 4d 1a 4c 2b 2c
     5P 1a 4a 2a 4b 6a 4c 3a 4d 2b 2c

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1 -1  1 -1  1  1
X.3      1  A -1 -A -1  A  1 -A  1 -1
X.4      1 -A -1  A -1 -A  1  A  1 -1
X.5      2  . -2  .  1  . -1  .  2 -2
X.6      2  .  2  . -1  . -1  .  2  2
X.7      3 -1 -1 -1  .  1  .  1 -1  3
X.8      3  1 -1  1  . -1  . -1 -1  3
X.9      3  A  1 -A  . -A  .  A -1 -3
X.10     3 -A  1  A  .  A  . -A -1 -3

A = -E(4)
  = -Sqrt(-1) = -i