Properties

Label 12T123
Order \(240\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2\times S_5$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $123$
Group :  $C_2\times S_5$
CHM label :  $L(6):2[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5,7,9)(2,4,6,8,12), (1,11)(2,4)(3,5)(6,8)(7,9)(10,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $\PGL(2,5)$

Low degree siblings

10T22 x 2, 12T123, 20T62 x 2, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 x 2

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 $ $1$ $1$ $()$
$ 4, 4, 1, 1, 1, 1 $ $30$ $4$ $( 4, 6, 8,10)( 5, 7, 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 5, 5, 1, 1 $ $24$ $5$ $( 2, 4,10, 6, 8)( 3, 5,11, 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6,11)( 7,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3,12)( 4, 7)( 5, 6)( 8,11)( 9,10)$
$ 6, 6 $ $20$ $6$ $( 1, 2, 5,12, 3, 4)( 6,11, 8, 7,10, 9)$
$ 10, 2 $ $24$ $10$ $( 1, 2, 5, 6, 9,12, 3, 4, 7, 8)(10,11)$
$ 6, 6 $ $20$ $6$ $( 1, 2, 5, 8,11, 6)( 3, 4, 9,10, 7,12)$
$ 4, 4, 2, 2 $ $30$ $4$ $( 1, 2, 5,10)( 3, 4,11,12)( 6, 7)( 8, 9)$
$ 6, 6 $ $20$ $6$ $( 1, 3, 5, 9,11, 7)( 2, 4, 8,10, 6,12)$
$ 3, 3, 3, 3 $ $20$ $3$ $( 1, 3, 5)( 2, 4,12)( 6,10, 8)( 7,11, 9)$
$ 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [240, 189]
Character table:   
      2  4  3  4  1  4  3  2   1  2  3  2  2  3  4
      3  1  .  .  .  .  1  1   .  1  .  1  1  1  1
      5  1  .  .  1  .  .  .   1  .  .  .  .  .  1

        1a 4a 2a 5a 2b 2c 6a 10a 6b 4b 6c 3a 2d 2e
     2P 1a 2a 1a 5a 1a 1a 3a  5a 3a 2a 3a 3a 1a 1a
     3P 1a 4a 2a 5a 2b 2c 2e 10a 2c 4b 2d 1a 2d 2e
     5P 1a 4a 2a 1a 2b 2c 6a  2e 6b 4b 6c 3a 2d 2e
     7P 1a 4a 2a 5a 2b 2c 6a 10a 6b 4b 6c 3a 2d 2e

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