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Properties

Label 12T124
Order \(240\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5:C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $124$
Group :  $A_5:C_4$
CHM label :  $[2]L(6):2_{12}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10)(3,5,7,9,11), (1,3,12,2)(4,6,5,7)(8,11,9,10), (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:  $C_2$
4:  $C_4$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

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

Low degree siblings

12T124, 20T66, 24T571, 24T578, 40T178, 40T179

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $15$ $2$ $( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 1, 1 $ $30$ $4$ $( 2, 3)( 4, 7,10, 9)( 5, 6,11, 8)$
$ 4, 4, 2, 1, 1 $ $30$ $4$ $( 2, 3)( 4, 9,10, 7)( 5, 8,11, 6)$
$ 5, 5, 1, 1 $ $24$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$
$ 12 $ $20$ $12$ $( 1, 2, 4,10, 7, 8,12, 3, 5,11, 6, 9)$
$ 6, 6 $ $20$ $6$ $( 1, 2, 5,12, 3, 4)( 6,11, 9, 7,10, 8)$
$ 10, 2 $ $24$ $10$ $( 1, 2, 6, 5, 9,12, 3, 7, 4, 8)(10,11)$
$ 3, 3, 3, 3 $ $20$ $3$ $( 1, 2, 6)( 3, 7,12)( 4,11, 9)( 5,10, 8)$
$ 12 $ $20$ $12$ $( 1, 2, 7, 8,10, 4,12, 3, 6, 9,11, 5)$
$ 4, 4, 4 $ $10$ $4$ $( 1, 2,12, 3)( 4, 7, 5, 6)( 8,10, 9,11)$
$ 4, 4, 4 $ $10$ $4$ $( 1, 2,12, 3)( 4, 9, 5, 8)( 6,10, 7,11)$
$ 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, 91]
Character table:    
      2  4  4  3  3  1  4   2  2   1  2   2  3  3  4
      3  1  .  .  .  .  .   1  1   .  1   1  1  1  1
      5  1  .  .  .  1  .   .  .   1  .   .  .  .  1

        1a 2a 4a 4b 5a 2b 12a 6a 10a 3a 12b 4c 4d 2c
     2P 1a 1a 2a 2a 5a 1a  6a 3a  5a 3a  6a 2c 2c 1a
     3P 1a 2a 4b 4a 5a 2b  4d 2c 10a 1a  4c 4d 4c 2c
     5P 1a 2a 4a 4b 1a 2b 12a 6a  2c 3a 12b 4c 4d 2c
     7P 1a 2a 4b 4a 5a 2b 12b 6a 10a 3a 12a 4d 4c 2c
    11P 1a 2a 4b 4a 5a 2b 12b 6a 10a 3a 12a 4d 4c 2c

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  A -A  1  1  -A -1  -1  1   A -A  A -1
X.4      1 -1 -A  A  1  1   A -1  -1  1  -A  A -A -1
X.5      4  .  .  . -1  .   1  1  -1  1   1 -2 -2  4
X.6      4  .  .  . -1  .  -1  1  -1  1  -1  2  2  4
X.7      4  .  .  . -1  .   A -1   1  1  -A  B -B -4
X.8      4  .  .  . -1  .  -A -1   1  1   A -B  B -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 -A  A  .  1  -A  1   . -1   A -A  A -5
X.12     5 -1  A -A  .  1   A  1   . -1  -A  A -A -5
X.13     6 -2  .  .  1 -2   .  .   1  .   .  .  .  6
X.14     6  2  .  .  1 -2   .  .  -1  .   .  .  . -6

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