Properties

Label 24T10
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $10$
Group :  $S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,24), (1,5,12,24)(2,6,11,23)(3,7,14,20)(4,8,13,19)(9,16,21,17)(10,15,22,18)
$|\Aut(F/K)|$:  $24$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_3$, $S_4$, $S_4$

Degree 8: $S_4$

Degree 12: $S_4$, $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T8, 12T9

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,21)( 4,22)( 5, 7)( 6, 8)( 9,19)(10,20)(11,15)(12,16)(13,24)(14,23) (17,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 4,21)( 2, 3,22)( 5,11,19)( 6,12,20)( 7, 9,15)( 8,10,16)(13,23,18) (14,24,17)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 5,12,24)( 2, 6,11,23)( 3, 7,14,20)( 4, 8,13,19)( 9,16,21,17)(10,15,22,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,12)( 2,11)( 3,14)( 4,13)( 5,24)( 6,23)( 7,20)( 8,19)( 9,21)(10,22)(15,18) (16,17)$

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 12]
Character table:   
     2  3  2  .  2  3
     3  1  .  1  .  .

       1a 2a 3a 4a 2b
    2P 1a 1a 3a 2b 1a
    3P 1a 2a 1a 4a 2b

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