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Properties

Label 6T6
Order \(24\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $6$
Group :  $A_4\times C_2$
CHM label :  $2A_{4}(6) = [2^{3}]3 = 2 wr 3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5)(2,4,6), (3,6)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Low degree siblings

8T13, 12T6, 12T7, 24T9

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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1 $ $3$ $2$ $(3,6)$
$ 2, 2, 1, 1 $ $3$ $2$ $(2,5)(3,6)$
$ 3, 3 $ $4$ $3$ $(1,2,3)(4,5,6)$
$ 6 $ $4$ $6$ $(1,2,3,4,5,6)$
$ 3, 3 $ $4$ $3$ $(1,3,2)(4,6,5)$
$ 6 $ $4$ $6$ $(1,3,5,4,6,2)$
$ 2, 2, 2 $ $1$ $2$ $(1,4)(2,5)(3,6)$

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 13]
Character table:    
     2  3  3  3  1   1  1   1  3
     3  1  .  .  1   1  1   1  1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3