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Properties

Label 12T52
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\GL(2,Z/4)$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $52$
Group :  $\GL(2,Z/4)$
CHM label :  $1/2c[1/16.D(4)^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,9)(6,12), (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,11)(2,10)(3,9)(4,8)(5,7), (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:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $S_4$, $(C_6\times C_2):C_2$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Low degree siblings

12T49 x 2, 12T50, 16T186, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 195]
Character table:    
      2  5  5  3  3  5  3  2  2  3  2  2  4  4  5
      3  1  .  .  .  .  .  1  1  .  1  1  .  1  1

        1a 2a 2b 4a 2c 4b 6a 6b 4c 6c 3a 2d 2e 2f
     2P 1a 1a 1a 2c 1a 2a 3a 3a 2f 3a 3a 1a 1a 1a
     3P 1a 2a 2b 4a 2c 4b 2e 2e 4c 2f 1a 2d 2e 2f
     5P 1a 2a 2b 4a 2c 4b 6b 6a 4c 6c 3a 2d 2e 2f

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1  1 -1 -1  1 -1  1  1 -1  1  1  1  1  1
X.3      1  1 -1 -1  1  1 -1 -1  1  1  1 -1 -1  1
X.4      1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1
X.5      2  2  .  .  2  .  1  1  . -1 -1 -2 -2  2
X.6      2  2  .  .  2  . -1 -1  . -1 -1  2  2  2
X.7      2 -2  .  .  2  .  .  .  . -2  2  .  . -2
X.8      2 -2  .  .  2  .  A -A  .  1 -1  .  . -2
X.9      2 -2  .  .  2  . -A  A  .  1 -1  .  . -2
X.10     3 -1 -1  1 -1 -1  .  .  1  .  .  1 -3  3
X.11     3 -1 -1  1 -1  1  .  . -1  .  . -1  3  3
X.12     3 -1  1 -1 -1 -1  .  .  1  .  . -1  3  3
X.13     3 -1  1 -1 -1  1  .  . -1  .  .  1 -3  3
X.14     6  2  .  . -2  .  .  .  .  .  .  .  . -6

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