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Properties

Label 12T15
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_6\times C_2):C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $15$
Group :  $(C_6\times C_2):C_2$
CHM label :  $1/2[3:2]dD(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,8)(4,10)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Low degree siblings

12T13, 24T14

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 8]
Character table:    
     2  3  2  2  2  2  2  2  3  2
     3  1  1  .  .  1  1  1  1  1

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

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

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3