# Properties

 Label 40T62 Degree $40$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

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Show commands: Magma

magma: G := TransitiveGroup(40, 62);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $62$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); 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) magma: Generators(G);

## 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 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, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 120.34 magma: IdentifyGroup(G);
 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 

magma: CharacterTable(G);