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Properties

Label 40T62
Order \(120\)
n \(40\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $62$
Group :  $S_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29,19,2,30,20)(3,32,17,4,31,18)(5,25,35,13,37,23)(6,26,36,14,38,24)(7,27,33,16,39,22)(8,28,34,15,40,21)(9,12)(10,11), (1,18,14)(2,17,13)(3,20,15)(4,19,16)(9,39,33)(10,40,34)(11,38,36)(12,37,35)(21,25,31)(22,26,32)(23,28,29)(24,27,30)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $S_5$

Degree 8: None

Degree 10: $S_5$, $S_5$

Degree 20: 20T30, 20T32, 20T35

Low degree siblings

5T5, 6T14, 10T12, 10T13

Siblings are shown with degree $\leq 10$

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $20$ $3$ $( 5,14,18)( 6,13,17)( 7,15,20)( 8,16,19)( 9,21,36)(10,22,35)(11,23,33) (12,24,34)(25,29,38)(26,30,37)(27,32,40)(28,31,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,34)(10,33)(11,35)(12,36)(13,18)(14,17)(15,19) (16,20)(21,24)(22,23)(25,30)(26,29)(27,31)(28,32)(37,38)(39,40)$
$ 6, 6, 6, 6, 6, 6, 2, 2 $ $20$ $6$ $( 1, 3)( 2, 4)( 5,25,18,38,14,29)( 6,26,17,37,13,30)( 7,27,20,40,15,32) ( 8,28,19,39,16,31)( 9,22,36,10,21,35)(11,24,33,12,23,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 4)( 2, 3)( 5,26)( 6,25)( 7,28)( 8,27)( 9,11)(10,12)(13,38)(14,37)(15,39) (16,40)(17,29)(18,30)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 6,14,17)( 2, 5,13,18)( 3, 8,15,19)( 4, 7,16,20)( 9,22,31,37)(10,21,32,38) (11,24,29,40)(12,23,30,39)(25,34,28,35)(26,33,27,36)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 8,12,24,30)( 2, 7,11,23,29)( 3, 6, 9,21,31)( 4, 5,10,22,32) (13,33,28,20,38)(14,34,27,19,37)(15,36,25,17,39)(16,35,26,18,40)$

Group invariants

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

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

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