# Properties

 Label 41T3 Degree $41$ Order $164$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{41}:C_{4}$

Show commands: Magma

magma: G := TransitiveGroup(41, 3);

## Group action invariants

 Degree $n$: $41$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $3$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{41}:C_{4}$ 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: (1,32,40,9)(2,23,39,18)(3,14,38,27)(4,5,37,36)(6,28,35,13)(7,19,34,22)(8,10,33,31)(11,24,30,17)(12,15,29,26)(16,20,25,21), (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)
$2$:  $C_2$
$4$:  $C_4$

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 $1^{41}$ $1$ $1$ $()$ 2A $2^{20},1$ $41$ $2$ $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$ 4A1 $4^{10},1$ $41$ $4$ $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23)( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$ 4A-1 $4^{10},1$ $41$ $4$ $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20)( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$ 41A1 $41$ $4$ $41$ $( 1,21,41,20,40,19,39,18,38,17,37,16,36,15,35,14,34,13,33,12,32,11,31,10,30, 9,29, 8,28, 7,27, 6,26, 5,25, 4,24, 3,23, 2,22)$ 41A2 $41$ $4$ $41$ $( 1, 8,15,22,29,36, 2, 9,16,23,30,37, 3,10,17,24,31,38, 4,11,18,25,32,39, 5,12,19,26,33,40, 6,13,20,27,34,41, 7,14,21,28,35)$ 41A3 $41$ $4$ $41$ $( 1,31,20, 9,39,28,17, 6,36,25,14, 3,33,22,11,41,30,19, 8,38,27,16, 5,35,24,13, 2,32,21,10,40,29,18, 7,37,26,15, 4,34,23,12)$ 41A4 $41$ $4$ $41$ $( 1,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ 41A6 $41$ $4$ $41$ $( 1,24, 6,29,11,34,16,39,21, 3,26, 8,31,13,36,18,41,23, 5,28,10,33,15,38,20, 2,25, 7,30,12,35,17,40,22, 4,27, 9,32,14,37,19)$ 41A7 $41$ $4$ $41$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$ 41A8 $41$ $4$ $41$ $( 1,29,16, 3,31,18, 5,33,20, 7,35,22, 9,37,24,11,39,26,13,41,28,15, 2,30,17, 4,32,19, 6,34,21, 8,36,23,10,38,25,12,40,27,14)$ 41A11 $41$ $4$ $41$ $( 1,11,21,31,41,10,20,30,40, 9,19,29,39, 8,18,28,38, 7,17,27,37, 6,16,26,36, 5,15,25,35, 4,14,24,34, 3,13,23,33, 2,12,22,32)$ 41A12 $41$ $4$ $41$ $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$ 41A16 $41$ $4$ $41$ $( 1,39,36,33,30,27,24,21,18,15,12, 9, 6, 3,41,38,35,32,29,26,23,20,17,14,11, 8, 5, 2,40,37,34,31,28,25,22,19,16,13,10, 7, 4)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $164=2^{2} \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 164.3 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 41A1 41A2 41A3 41A4 41A6 41A7 41A8 41A11 41A12 41A16 Size 1 41 41 41 4 4 4 4 4 4 4 4 4 4 2 P 1A 1A 2A 2A 41A16 41A7 41A11 41A1 41A2 41A4 41A6 41A8 41A12 41A3 41 P 1A 2A 4A1 4A-1 41A1 41A3 41A7 41A2 41A4 41A8 41A12 41A16 41A11 41A6 Type 164.3.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 164.3.1b R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 164.3.1c1 C $1$ $−1$ $−i$ $i$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 164.3.1c2 C $1$ $−1$ $i$ $−i$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 164.3.4a1 R $4$ $0$ $0$ $0$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−10+ζ41−8+ζ418+ζ4110$ 164.3.4a2 R $4$ $0$ $0$ $0$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ 164.3.4a3 R $4$ $0$ $0$ $0$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−13+ζ41−6+ζ416+ζ4113$ 164.3.4a4 R $4$ $0$ $0$ $0$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ 164.3.4a5 R $4$ $0$ $0$ $0$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−9+ζ41−1+ζ41+ζ419$ 164.3.4a6 R $4$ $0$ $0$ $0$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−14+ζ41−3+ζ413+ζ4114$ 164.3.4a7 R $4$ $0$ $0$ $0$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−19+ζ41−7+ζ417+ζ4119$ 164.3.4a8 R $4$ $0$ $0$ $0$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−5+ζ41−4+ζ414+ζ415$ 164.3.4a9 R $4$ $0$ $0$ $0$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−18+ζ41−2+ζ412+ζ4118$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ 164.3.4a10 R $4$ $0$ $0$ $0$ $ζ41−5+ζ41−4+ζ414+ζ415$ $ζ41−10+ζ41−8+ζ418+ζ4110$ $ζ41−15+ζ41−12+ζ4112+ζ4115$ $ζ41−20+ζ41−16+ζ4116+ζ4120$ $ζ41−17+ζ41−11+ζ4111+ζ4117$ $ζ41−13+ζ41−6+ζ416+ζ4113$ $ζ41−9+ζ41−1+ζ41+ζ419$ $ζ41−14+ζ41−3+ζ413+ζ4114$ $ζ41−19+ζ41−7+ζ417+ζ4119$ $ζ41−18+ζ41−2+ζ412+ζ4118$

magma: CharacterTable(G);