# Properties

 Label 20T31 Degree $20$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $C_2\times A_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 31);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $31$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2\times A_5$ 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: (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16), (1,12,3,15,5,2,11,4,16,6)(7,10,17,20,13,8,9,18,19,14) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

## Low degree siblings

10T11, 12T75, 12T76, 20T36, 24T203, 30T29, 30T30, 40T61

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 $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ 2B $2^{10}$ $15$ $2$ $10$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$ 2C $2^{8},1^{4}$ $15$ $2$ $8$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,13)( 6,14)( 7,11)( 8,12)$ 3A $3^{6},1^{2}$ $20$ $3$ $12$ $( 1, 7, 9)( 2, 8,10)( 5,17,19)( 6,18,20)(11,16,13)(12,15,14)$ 5A1 $5^{4}$ $12$ $5$ $16$ $( 1,19, 5,11, 9)( 2,20, 6,12,10)( 3,13,17, 7,16)( 4,14,18, 8,15)$ 5A2 $5^{4}$ $12$ $5$ $16$ $( 1, 5, 9,19,11)( 2, 6,10,20,12)( 3,17,16,13, 7)( 4,18,15,14, 8)$ 6A $6^{3},2$ $20$ $6$ $16$ $( 1, 8, 9, 2, 7,10)( 3, 4)( 5,18,19, 6,17,20)(11,15,13,12,16,14)$ 10A1 $10^{2}$ $12$ $10$ $18$ $( 1, 6, 9,20,11, 2, 5,10,19,12)( 3,18,16,14, 7, 4,17,15,13, 8)$ 10A3 $10^{2}$ $12$ $10$ $18$ $( 1,20, 5,12, 9, 2,19, 6,11,10)( 3,14,17, 8,16, 4,13,18, 7,15)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/8$

## 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: 120.35 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3 Size 1 1 15 15 20 12 12 20 12 12 2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2 3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1 5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A Type 120.35.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 120.35.1b R $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ 120.35.3a1 R $3$ $3$ $−1$ $−1$ $0$ $−ζ5−1−ζ5$ $−ζ5−2−ζ52$ $0$ $−ζ5−2−ζ52$ $−ζ5−1−ζ5$ 120.35.3a2 R $3$ $3$ $−1$ $−1$ $0$ $−ζ5−2−ζ52$ $−ζ5−1−ζ5$ $0$ $−ζ5−1−ζ5$ $−ζ5−2−ζ52$ 120.35.3b1 R $3$ $−3$ $1$ $−1$ $0$ $−ζ5−1−ζ5$ $−ζ5−2−ζ52$ $0$ $ζ5−2+ζ52$ $ζ5−1+ζ5$ 120.35.3b2 R $3$ $−3$ $1$ $−1$ $0$ $−ζ5−2−ζ52$ $−ζ5−1−ζ5$ $0$ $ζ5−1+ζ5$ $ζ5−2+ζ52$ 120.35.4a R $4$ $4$ $0$ $0$ $1$ $−1$ $−1$ $1$ $−1$ $−1$ 120.35.4b R $4$ $−4$ $0$ $0$ $1$ $−1$ $−1$ $−1$ $1$ $1$ 120.35.5a R $5$ $5$ $1$ $1$ $−1$ $0$ $0$ $−1$ $0$ $0$ 120.35.5b R $5$ $−5$ $−1$ $1$ $−1$ $0$ $0$ $1$ $0$ $0$

magma: CharacterTable(G);