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Properties

Label 24T202
Order \(120\)
n \(24\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $202$
Group :  $S_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,4)(5,9)(6,10)(7,13)(8,14)(11,19)(12,20)(15,17)(16,18)(21,23)(22,24), (3,17,11,7,5)(4,18,12,8,6)(9,14,22,20,15)(10,13,21,19,16)
$|\Aut(F/K)|$:  $4$

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: $\PGL(2,5)$

Degree 8: None

Degree 12: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1 $ $24$ $5$ $( 3, 5, 7,11,17)( 4, 6, 8,12,18)( 9,15,20,22,14)(10,16,19,21,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,12)( 8,11)( 9,16)(10,15)(13,20)(14,19)(21,22) (23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,13)( 8,14)(11,19)(12,20)(15,17)(16,18)(21,23) (22,24)$
$ 4, 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 3, 9,17)( 2, 4,10,18)( 5,13, 6,14)( 7,19,23,21)( 8,20,24,22)(11,15,12,16)$
$ 6, 6, 6, 6 $ $20$ $6$ $( 1, 3,13,18,10,11)( 2, 4,14,17, 9,12)( 5,19, 7,15,24,22)( 6,20, 8,16,23,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 9,20)( 2,10,19)( 3,12, 5)( 4,11, 6)( 7,17,23)( 8,18,24)(13,21,15) (14,22,16)$

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  2  1  1
     3  1  .  .  1  .  1  1
     5  1  1  .  .  .  .  .

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