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Properties

Label 12T37
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $37$
Group :  $C_2\times S_3^2$
CHM label :  $[3^{2}:2]E(4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5)(2,10)(4,8)(7,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$ x 2
8:  $C_2^3$
12:  $D_{6}$ x 6
24:  $S_3 \times C_2^2$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Low degree siblings

12T37, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 4

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 46]
Character table:    
      2  3  3  1  3  3  2  2  3  2  1  2  3  2  2  3  2  2  3
      3  2  .  2  1  1  1  1  .  2  2  2  1  1  1  1  2  2  2

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

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