# Properties

 Label 41T4 Degree $41$ Order $205$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{41}:C_{5}$

Show commands: Magma

magma: G := TransitiveGroup(41, 4);

## Group action invariants

 Degree $n$: $41$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{41}:C_{5}$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,10,18,16,37)(2,20,36,32,33)(3,30,13,7,29)(4,40,31,23,25)(5,9,8,39,21)(6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

There are no siblings 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 $1^{41}$ $1$ $1$ $()$ 5A1 $5^{8},1$ $41$ $5$ $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24)( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$ 5A-1 $5^{8},1$ $41$ $5$ $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32)( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$ 5A2 $5^{8},1$ $41$ $5$ $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26)( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$ 5A-2 $5^{8},1$ $41$ $5$ $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41)( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$ 41A1 $41$ $5$ $41$ $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$ 41A-1 $41$ $5$ $41$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$ 41A2 $41$ $5$ $41$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ 41A-2 $41$ $5$ $41$ $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15,19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$ 41A3 $41$ $5$ $41$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$ 41A-3 $41$ $5$ $41$ $( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19,30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$ 41A6 $41$ $5$ $41$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ 41A-6 $41$ $5$ $41$ $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22,28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $205=5 \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 205.1 magma: IdentifyGroup(G); Character table:

 1A 5A1 5A-1 5A2 5A-2 41A1 41A-1 41A2 41A-2 41A3 41A-3 41A6 41A-6 Size 1 41 41 41 41 5 5 5 5 5 5 5 5 5 P 1A 5A-1 5A1 5A2 5A-2 41A3 41A6 41A-1 41A-2 41A1 41A-6 41A2 41A-3 41 P 1A 1A 1A 1A 1A 41A-3 41A-6 41A1 41A2 41A-1 41A6 41A-2 41A3 Type 205.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 205.1.1b1 C $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 205.1.1b2 C $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 205.1.1b3 C $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 205.1.1b4 C $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 205.1.5a1 C $5$ $0$ $0$ $0$ $0$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ 205.1.5a2 C $5$ $0$ $0$ $0$ $0$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ 205.1.5a3 C $5$ $0$ $0$ $0$ $0$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ 205.1.5a4 C $5$ $0$ $0$ $0$ $0$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ 205.1.5a5 C $5$ $0$ $0$ $0$ $0$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ 205.1.5a6 C $5$ $0$ $0$ $0$ $0$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ 205.1.5a7 C $5$ $0$ $0$ $0$ $0$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ 205.1.5a8 C $5$ $0$ $0$ $0$ $0$ $ζ41−20+ζ41−2+ζ415+ζ418+ζ419$ $ζ41−9+ζ41−8+ζ41−5+ζ412+ζ4120$ $ζ41−4+ζ41+ζ4110+ζ4116+ζ4118$ $ζ41−18+ζ41−16+ζ41−10+ζ41−1+ζ414$ $ζ41−19+ζ41−17+ζ41−14+ζ41−6+ζ4115$ $ζ41−15+ζ416+ζ4114+ζ4117+ζ4119$ $ζ41−12+ζ41−11+ζ413+ζ417+ζ4113$ $ζ41−13+ζ41−7+ζ41−3+ζ4111+ζ4112$

magma: CharacterTable(G);