Properties

Label 12T173
Order \(648\)
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 $t$:  $173$
CHM label:  $[3^{4}:2]4_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,10)(4,8)(7,11), (4,8,12), (1,4,7,10,5,8,11,2)(3,6,9,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$
72:  $C_3^2:C_8$ x 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

12T173 x 7, 18T196 x 4, 24T1525 x 8, 36T1093 x 4, 36T1216 x 4, 36T1217 x 16, 36T1235 x 2, 36T1236 x 8

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

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [648, 712]
Character table:   
      2  3  .  .  .  .  .  .  .  .  .  .  3  3  3   3   3   3   3
      3  4  4  4  4  4  4  4  4  4  4  4  .  .  .   .   .   .   .

        1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 2a 4a 4b  8a  8b  8c  8d
     2P 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 1a 2a 2a  4a  4a  4b  4b
     3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a 4b 4a  8c  8d  8a  8b
     5P 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 2a 4a 4b  8b  8a  8d  8c
     7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 2a 4b 4a  8d  8c  8b  8a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)^3