# Properties

 Label 20T26 Degree $20$ Order $100$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 26);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $26$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5:F_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $5$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,15,20,10)(2,12,19,8)(3,14,18,6)(4,11,17,9)(5,13,16,7), (1,19,5,20,4,16,3,17,2,18)(6,11,9,13,7,15,10,12,8,14) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  $F_5$, 20T2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

## Low degree siblings

25T11

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 1, 1, 1, 1, 1$ $4$ $5$ $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)$ $5, 5, 5, 1, 1, 1, 1, 1$ $4$ $5$ $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)$ $5, 5, 5, 1, 1, 1, 1, 1$ $4$ $5$ $( 6, 9, 7,10, 8)(11,12,13,14,15)(16,20,19,18,17)$ $5, 5, 5, 1, 1, 1, 1, 1$ $4$ $5$ $( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)$ $5, 5, 5, 5$ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)$ $5, 5, 5, 5$ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 8,10, 7, 9)(11,14,12,15,13)(16,20,19,18,17)$ $5, 5, 5, 5$ $2$ $5$ $( 1, 3, 5, 2, 4)( 6,10, 9, 8, 7)(11,12,13,14,15)(16,19,17,20,18)$ $4, 4, 4, 4, 4$ $25$ $4$ $( 1, 6,16,15)( 2, 9,20,12)( 3, 7,19,14)( 4,10,18,11)( 5, 8,17,13)$ $4, 4, 4, 4, 4$ $25$ $4$ $( 1,11,17, 6)( 2,13,16, 9)( 3,15,20, 7)( 4,12,19,10)( 5,14,18, 8)$ $10, 10$ $10$ $10$ $( 1,16, 4,18, 2,20, 5,17, 3,19)( 6,11, 7,15, 8,14, 9,13,10,12)$ $10, 10$ $10$ $10$ $( 1,16, 2,20, 3,19, 4,18, 5,17)( 6,12, 8,15,10,13, 7,11, 9,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1,16)( 2,20)( 3,19)( 4,18)( 5,17)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$

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); Label: 100.10 magma: IdentifyGroup(G);
 Character table:  2 2 . . . . . 1 1 2 2 1 1 2 5 2 2 2 2 2 2 2 2 . . 1 1 1 1a 5a 5b 5c 5d 5e 5f 5g 4a 4b 10a 10b 2a 2P 1a 5b 5d 5a 5c 5e 5g 5f 2a 2a 5g 5f 1a 3P 1a 5c 5a 5d 5b 5e 5g 5f 4b 4a 10b 10a 2a 5P 1a 1a 1a 1a 1a 1a 1a 1a 4a 4b 2a 2a 2a 7P 1a 5b 5d 5a 5c 5e 5g 5f 4b 4a 10b 10a 2a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 1 -1 -1 1 1 1 X.3 1 1 1 1 1 1 1 1 E -E -1 -1 -1 X.4 1 1 1 1 1 1 1 1 -E E -1 -1 -1 X.5 2 A *A *A A 2 A *A . . -A -*A -2 X.6 2 *A A A *A 2 *A A . . -*A -A -2 X.7 2 A *A *A A 2 A *A . . A *A 2 X.8 2 *A A A *A 2 *A A . . *A A 2 X.9 4 -1 -1 -1 -1 -1 4 4 . . . . . X.10 4 B C /C /B -1 D *D . . . . . X.11 4 /B /C C B -1 D *D . . . . . X.12 4 C /B B /C -1 *D D . . . . . X.13 4 /C B /B C -1 *D D . . . . . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = -E(5)-E(5)^3+E(5)^4 C = -E(5)-E(5)^2+E(5)^3 D = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 E = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);