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Properties

Label 12T10
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3 \times C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $10$
Group :  $S_3 \times C_2^2$
CHM label :  $S(3)[x]E(4)$
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Low degree siblings

12T10 x 3, 24T11

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 14]
Character table:    
      2  3  3  3  2  2  3  3  3  2  3  2  3
      3  1  .  .  1  1  .  .  1  1  1  1  1

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  .  . -1 -1  .  .  2 -1  2 -1  2
X.10     2  .  . -1  1  .  . -2 -1 -2  1  2
X.11     2  .  .  1 -1  .  . -2 -1  2  1 -2
X.12     2  .  .  1  1  .  .  2 -1 -2 -1 -2