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Properties

Label 12T62
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 $t$ :  $62$
Group :  $C_4^2:C_3:C_2$
CHM label :  $[4^{2}]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,10)(4,8)(7,11), (1,10,7,4)(3,6,9,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

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

12T63, 12T64, 12T65, 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 $ $12$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3,12, 9, 6)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 9)( 4, 5,10,11)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 6, 9,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 64]
Character table:    
      2  5  3  4  5  .  3  3  3  5  5
      3  1  .  .  .  1  .  .  .  .  .

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

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

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