# Properties

 Label 20T27 Degree $20$ Order $100$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5:F_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 27);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $27$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5:F_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $10$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,12,18,7)(2,11,17,8)(3,5,15,14)(4,6,16,13)(9,20,10,19), (3,8,12,16,19)(4,7,11,15,20) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

## Low degree siblings

10T10 x 2, 20T27, 25T10

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 Representative 1A $1^{20}$ $1$ $1$ $()$ 2A $2^{10}$ $25$ $2$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$ 4A1 $4^{5}$ $25$ $4$ $( 1,12,18, 7)( 2,11,17, 8)( 3, 5,15,14)( 4, 6,16,13)( 9,20,10,19)$ 4A-1 $4^{5}$ $25$ $4$ $( 1, 7,18,12)( 2, 8,17,11)( 3,14,15, 5)( 4,13,16, 6)( 9,19,10,20)$ 5A $5^{4}$ $4$ $5$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,12,19, 8,16)( 4,11,20, 7,15)$ 5B $5^{4}$ $4$ $5$ $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3, 8,12,16,19)( 4, 7,11,15,20)$ 5C1 $5^{4}$ $4$ $5$ $( 1,17,14,10, 6)( 2,18,13, 9, 5)( 3,19,16,12, 8)( 4,20,15,11, 7)$ 5C2 $5^{2},1^{10}$ $4$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)$ 5D1 $5^{2},1^{10}$ $4$ $5$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$ 5D2 $5^{4}$ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 100.12 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 5A 5B 5C1 5C2 5D1 5D2 Size 1 25 25 25 4 4 4 4 4 4 2 P 1A 1A 2A 2A 5D1 5A 5D2 5C2 5C1 5B 5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A Type 100.12.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 100.12.1b R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ 100.12.1c1 C $1$ $−1$ $−i$ $i$ $1$ $1$ $1$ $1$ $1$ $1$ 100.12.1c2 C $1$ $−1$ $i$ $−i$ $1$ $1$ $1$ $1$ $1$ $1$ 100.12.4a R $4$ $0$ $0$ $0$ $−1$ $4$ $−1$ $−1$ $−1$ $−1$ 100.12.4b R $4$ $0$ $0$ $0$ $4$ $−1$ $−1$ $−1$ $−1$ $−1$ 100.12.4c1 R $4$ $0$ $0$ $0$ $−1$ $−1$ $2ζ5−2+2ζ52$ $2ζ5−1+2ζ5$ $ζ5−2+2+ζ52$ $−ζ5−2+1−ζ52$ 100.12.4c2 R $4$ $0$ $0$ $0$ $−1$ $−1$ $2ζ5−1+2ζ5$ $2ζ5−2+2ζ52$ $−ζ5−2+1−ζ52$ $ζ5−2+2+ζ52$ 100.12.4d1 R $4$ $0$ $0$ $0$ $−1$ $−1$ $ζ5−2+2+ζ52$ $−ζ5−2+1−ζ52$ $2ζ5−1+2ζ5$ $2ζ5−2+2ζ52$ 100.12.4d2 R $4$ $0$ $0$ $0$ $−1$ $−1$ $−ζ5−2+1−ζ52$ $ζ5−2+2+ζ52$ $2ζ5−2+2ζ52$ $2ζ5−1+2ζ5$

magma: CharacterTable(G);