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Properties

Label 12T61
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4:D_4:C_3$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $61$
Group :  $C_4:D_4:C_3$
CHM label :  $[2^{3}]A_{4}(6)_{4}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,8)(3,9)(4,11)(5,10)(6,7), (1,12)(2,3)(6,7)(8,9), (1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T60 x 2, 12T61, 16T185, 24T187 x 2, 24T188 x 2, 24T189, 24T190, 32T391

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 $ $4$ $2$ $( 4, 5)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 8)( 3, 9)( 4,11)( 5,10)( 6, 7)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 9,11)( 7, 8,10)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 2)( 3,12, 5)( 6,11, 8)( 7,10, 9)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 4, 3,12, 5, 2)( 6,10, 9)( 7,11, 8)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 72]
Character table:    
      2  5  3  5  4  3  1   1  1   1  4
      3  1  1  .  .  .  1   1  1   1  .

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

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

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