Properties

 Label 20T30 Degree $20$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 30);

Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $30$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3)(2,4)(5,13)(6,14)(15,17)(16,18)(19,20), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19) 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: None

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 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

 Label Cycle Type Size Order Index Representative 1A $1^{20}$ $1$ $1$ $0$ $()$ 2A $2^{7},1^{6}$ $10$ $2$ $7$ $( 1, 9)( 2,10)(11,14)(12,13)(15,16)(17,20)(18,19)$ 2B $2^{10}$ $15$ $2$ $10$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,16)( 6,15)(11,19)(12,20)(13,14)(17,18)$ 3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1,10,15)( 2, 9,16)( 3,18,19)( 4,17,20)( 7,12,13)( 8,11,14)$ 4A $4^{5}$ $30$ $4$ $15$ $( 1, 3, 7, 9)( 2, 4, 8,10)( 5,11,16,19)( 6,12,15,20)(13,17,14,18)$ 5A $5^{4}$ $24$ $5$ $16$ $( 1, 5,10,19,11)( 2, 6, 9,20,12)( 3,17,16,13, 8)( 4,18,15,14, 7)$ 6A $6^{2},3^{2},2$ $20$ $6$ $15$ $( 1,11,20, 9,14,17)( 2,12,19,10,13,18)( 3, 8, 5)( 4, 7, 6)(15,16)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/7$

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); Nilpotency class: not nilpotent Label: 120.34 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 3A 4A 5A 6A Size 1 10 15 20 30 24 20 2 P 1A 1A 1A 3A 2B 5A 3A 3 P 1A 2A 2B 1A 4A 5A 2A 5 P 1A 2A 2B 3A 4A 1A 6A Type 120.34.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ 120.34.1b R $1$ $−1$ $1$ $1$ $−1$ $1$ $−1$ 120.34.4a R $4$ $2$ $0$ $1$ $0$ $−1$ $−1$ 120.34.4b R $4$ $−2$ $0$ $1$ $0$ $−1$ $1$ 120.34.5a R $5$ $−1$ $1$ $−1$ $1$ $0$ $−1$ 120.34.5b R $5$ $1$ $1$ $−1$ $−1$ $0$ $1$ 120.34.6a R $6$ $0$ $−2$ $0$ $0$ $1$ $0$

magma: CharacterTable(G);