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Properties

Label 12T65
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4^2:C_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $65$
Group :  $C_4^2:C_3:C_2$
CHM label :  $[1/2[1/2.2^{2}]^{3}]S_{4}(6c)_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3)(6,7)(8,9), (1,5)(2,9)(3,8)(4,12)(6,11)(7,10), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T62, 12T63, 12T64, 16T195, 24T191, 24T192, 24T193, 24T194, 32T399

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, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 8, 2, 1, 1 $ $12$ $8$ $( 2, 4, 9,11, 3, 5, 8,10)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 9, 3, 8)( 4,11, 5,10)$
$ 8, 2, 1, 1 $ $12$ $8$ $( 2,10, 8, 5, 3,11, 9, 4)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 9)( 7, 8)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,12, 3)( 4,11)( 5,10)( 6, 9, 7, 8)$
$ 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, 64]
Character table:    
      2  5  5  3  5  5  3  3  .  3  4
      3  1  .  .  .  .  .  .  1  .  .

        1a 2a 8a 4a 4b 8b 2b 3a 4c 4d
     2P 1a 1a 4b 2a 2a 4a 1a 3a 2a 2a
     3P 1a 2a 8b 4b 4a 8a 2b 1a 4c 4d
     5P 1a 2a 8a 4a 4b 8b 2b 3a 4c 4d
     7P 1a 2a 8b 4b 4a 8a 2b 3a 4c 4d

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -1-2*E(4)
  = -1-2*Sqrt(-1) = -1-2i