Properties

Label 12T215
Order \(1296\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$:  $12$
Transitive number $t$:  $215$
CHM label:  $1/2[F_{36}^{2}]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,7,6)(2,9,8,11,10,5,4,3), (4,8,12), (2,10)(4,8), (1,7)(2,8,10,4)(3,9,11,5)(6,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
16:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T215, 18T288, 18T297 x 2, 24T2899 x 2, 24T2900 x 2, 24T2901 x 2, 24T2928 x 2, 24T2939 x 2, 36T2164, 36T2165, 36T2166, 36T2195 x 2, 36T2307 x 2, 36T2308 x 2, 36T2313, 36T2314, 36T2319

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $16$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$ 4, 4, 2, 2 $ $81$ $4$ $( 1, 7)( 2, 8,10, 4)( 3, 9,11, 5)( 6,12)$
$ 4, 4, 2, 2 $ $81$ $4$ $( 1,11, 9, 7)( 2,12,10, 4)( 3, 5)( 6, 8)$
$ 8, 4 $ $162$ $8$ $( 1,12, 7, 6)( 2, 9, 8,11,10, 5, 4, 3)$
$ 8, 4 $ $162$ $8$ $( 1, 6, 7,12)( 2,11, 4, 9,10, 3, 8, 5)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 6,10)( 8,12)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $72$ $6$ $( 3, 7,11)( 6,10)( 8,12)$
$ 3, 3, 2, 2, 1, 1 $ $72$ $6$ $( 1, 5, 9)( 3, 7,11)( 6,10)( 8,12)$
$ 4, 4, 2, 2 $ $162$ $4$ $( 1, 7)( 2, 8, 6, 4)( 3, 9,11, 5)(10,12)$
$ 8, 4 $ $162$ $8$ $( 1,12,11,10)( 2, 9, 8, 7, 6, 5, 4, 3)$
$ 8, 4 $ $162$ $8$ $( 1, 6, 3, 8)( 2,11, 4, 9,10, 7,12, 5)$

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1296, 3508]
Character table:   
      2  4  1  .  1  .  .  .  4  4  4  3  3  3  1  1  3  3  3
      3  4  4  4  4  4  4  4  .  .  .  .  .  2  2  2  .  .  .

        1a 3a 3b 3c 3d 3e 3f 2a 4a 4b 8a 8b 2b 6a 6b 4c 8c 8d
     2P 1a 3a 3b 3c 3d 3e 3f 1a 2a 2a 4a 4b 1a 3a 3c 2a 4b 4a
     3P 1a 1a 1a 1a 1a 1a 1a 2a 4b 4a 8b 8a 2b 2b 2b 4c 8d 8c
     5P 1a 3a 3b 3c 3d 3e 3f 2a 4a 4b 8a 8b 2b 6a 6b 4c 8c 8d
     7P 1a 3a 3b 3c 3d 3e 3f 2a 4b 4a 8b 8a 2b 6a 6b 4c 8d 8c

X.1      1  1  1  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 -1 -1  1  1
X.3      1  1  1  1  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 -1 -1 -1 -1
X.5      1  1  1  1  1  1  1  1 -1 -1  B -B -1 -1 -1  1 -B  B
X.6      1  1  1  1  1  1  1  1 -1 -1 -B  B -1 -1 -1  1  B -B
X.7      1  1  1  1  1  1  1  1 -1 -1  B -B  1  1  1 -1  B -B
X.8      1  1  1  1  1  1  1  1 -1 -1 -B  B  1  1  1 -1 -B  B
X.9      2  2  2  2  2  2  2 -2  A -A  .  .  .  .  .  .  .  .
X.10     2  2  2  2  2  2  2 -2 -A  A  .  .  .  .  .  .  .  .
X.11     8  5  2  2 -1 -1 -4  .  .  .  .  . -4 -1  2  .  .  .
X.12     8  5  2  2 -1 -1 -4  .  .  .  .  .  4  1 -2  .  .  .
X.13     8  2 -4  5 -1 -1  2  .  .  .  .  . -4  2 -1  .  .  .
X.14     8  2 -4  5 -1 -1  2  .  .  .  .  .  4 -2  1  .  .  .
X.15    16 -8  4  4 -2 -2  1  .  .  .  .  .  .  .  .  .  .  .
X.16    16 -2 -2 -2 -2  7 -2  .  .  .  .  .  .  .  .  .  .  .
X.17    16 -2 -2 -2  7 -2 -2  .  .  .  .  .  .  .  .  .  .  .
X.18    16  4  1 -8 -2 -2  4  .  .  .  .  .  .  .  .  .  .  .

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