Properties

Label 12T74
Order \(120\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $74$
Group :  $S_5$
CHM label :  $S_{5}(12)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10)(3,5,7,9,11), (1,10)(2,7)(3,12)(4,5)(6,11)(8,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

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

5T5, 6T14, 10T12, 10T13, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [120, 34]
Character table:   
     2  3  3  .  2  1  2  1
     3  1  .  .  1  1  .  1
     5  1  .  1  .  .  .  .

       1a 2a 5a 2b 6a 4a 3a
    2P 1a 1a 5a 1a 3a 2a 3a
    3P 1a 2a 5a 2b 2b 4a 1a
    5P 1a 2a 1a 2b 6a 4a 3a

X.1     1  1  1  1  1  1  1
X.2     1  1  1 -1 -1 -1  1
X.3     4  . -1 -2  1  .  1
X.4     4  . -1  2 -1  .  1
X.5     5  1  .  1  1 -1 -1
X.6     5  1  . -1 -1  1 -1
X.7     6 -2  1  .  .  .  .