Properties

Label 12T30
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 $t$ :  $30$
Group :  $A_4:C_4$
CHM label :  $1/2[1/4.4^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,9)(6,12), (1,8,7,2)(3,6,9,12)(4,11,10,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

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: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T27, 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 2, 2, 2, 2 $ $6$ $4$ $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$
$ 4, 2, 2, 2, 2 $ $6$ $4$ $( 1, 2)( 3,12, 9, 6)( 4, 5)( 7, 8)(10,11)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 6, 6 $ $8$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 3,11)( 2, 4, 6)( 5, 7, 9)( 8,10,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

Group invariants

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

        1a 2a 2b 4a 4b 4c 4d 6a 3a 2c
     2P 1a 1a 1a 2a 2a 2c 2c 3a 3a 1a
     3P 1a 2a 2b 4b 4a 4d 4c 2c 1a 2c
     5P 1a 2a 2b 4a 4b 4c 4d 6a 3a 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 -1  1  A -A -A  A -1  1 -1
X.4      1 -1  1 -A  A  A -A -1  1 -1
X.5      2 -2  2  .  .  .  .  1 -1 -2
X.6      2  2  2  .  .  .  . -1 -1  2
X.7      3 -1 -1 -1 -1  1  1  .  .  3
X.8      3 -1 -1  1  1 -1 -1  .  .  3
X.9      3  1 -1  A -A  A -A  .  . -3
X.10     3  1 -1 -A  A -A  A  .  . -3

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