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Properties

Label 12T5
Order \(12\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3 : C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $5$
Group :  $C_3 : C_4$
CHM label :  $1/2[3:2]4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,7,2)(3,6,9,12)(4,11,10,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Low degree siblings

There are no siblings 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, 4 $ $3$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

Group invariants

Order:  $12=2^{2} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [12, 1]
Character table:    
     2  2  2  1  2  1  2
     3  1  .  1  .  1  1

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

X.1     1  1  1  1  1  1
X.2     1 -1  1 -1  1  1
X.3     1  A -1 -A  1 -1
X.4     1 -A -1  A  1 -1
X.5     2  .  1  . -1 -2
X.6     2  . -1  . -1  2

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