Properties

 Label 41T2 Degree $41$ Order $82$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $D_{41}$

Show commands: Magma

magma: G := TransitiveGroup(41, 2);

Group action invariants

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

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

magma: ConjugacyClasses(G);

Group invariants

 Order: $82=2 \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 82.1 magma: IdentifyGroup(G); Character table:

 1A 2A 41A1 41A2 41A3 41A4 41A5 41A6 41A7 41A8 41A9 41A10 41A11 41A12 41A13 41A14 41A15 41A16 41A17 41A18 41A19 41A20 Size 1 41 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P 1A 1A 41A15 41A1 41A4 41A9 41A5 41A10 41A17 41A12 41A7 41A11 41A6 41A8 41A13 41A18 41A14 41A19 41A16 41A3 41A2 41A20 41 P 1A 2A 41A17 41A18 41A10 41A2 41A8 41A16 41A19 41A11 41A3 41A7 41A15 41A20 41A12 41A4 41A6 41A14 41A1 41A13 41A5 41A9 Type 82.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 82.1.1b R $1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 82.1.2a1 R $2$ $0$ $ζ41−20+ζ4120$ $ζ41−1+ζ41$ $ζ41−19+ζ4119$ $ζ41−2+ζ412$ $ζ41−18+ζ4118$ $ζ41−3+ζ413$ $ζ41−17+ζ4117$ $ζ41−4+ζ414$ $ζ41−16+ζ4116$ $ζ41−5+ζ415$ $ζ41−15+ζ4115$ $ζ41−6+ζ416$ $ζ41−14+ζ4114$ $ζ41−7+ζ417$ $ζ41−13+ζ4113$ $ζ41−8+ζ418$ $ζ41−12+ζ4112$ $ζ41−9+ζ419$ $ζ41−11+ζ4111$ $ζ41−10+ζ4110$ 82.1.2a2 R $2$ $0$ $ζ41−19+ζ4119$ $ζ41−3+ζ413$ $ζ41−16+ζ4116$ $ζ41−6+ζ416$ $ζ41−13+ζ4113$ $ζ41−9+ζ419$ $ζ41−10+ζ4110$ $ζ41−12+ζ4112$ $ζ41−7+ζ417$ $ζ41−15+ζ4115$ $ζ41−4+ζ414$ $ζ41−18+ζ4118$ $ζ41−1+ζ41$ $ζ41−20+ζ4120$ $ζ41−2+ζ412$ $ζ41−17+ζ4117$ $ζ41−5+ζ415$ $ζ41−14+ζ4114$ $ζ41−8+ζ418$ $ζ41−11+ζ4111$ 82.1.2a3 R $2$ $0$ $ζ41−18+ζ4118$ $ζ41−5+ζ415$ $ζ41−13+ζ4113$ $ζ41−10+ζ4110$ $ζ41−8+ζ418$ $ζ41−15+ζ4115$ $ζ41−3+ζ413$ $ζ41−20+ζ4120$ $ζ41−2+ζ412$ $ζ41−16+ζ4116$ $ζ41−7+ζ417$ $ζ41−11+ζ4111$ $ζ41−12+ζ4112$ $ζ41−6+ζ416$ $ζ41−17+ζ4117$ $ζ41−1+ζ41$ $ζ41−19+ζ4119$ $ζ41−4+ζ414$ $ζ41−14+ζ4114$ $ζ41−9+ζ419$ 82.1.2a4 R $2$ $0$ $ζ41−17+ζ4117$ $ζ41−7+ζ417$ $ζ41−10+ζ4110$ $ζ41−14+ζ4114$ $ζ41−3+ζ413$ $ζ41−20+ζ4120$ $ζ41−4+ζ414$ $ζ41−13+ζ4113$ $ζ41−11+ζ4111$ $ζ41−6+ζ416$ $ζ41−18+ζ4118$ $ζ41−1+ζ41$ $ζ41−16+ζ4116$ $ζ41−8+ζ418$ $ζ41−9+ζ419$ $ζ41−15+ζ4115$ $ζ41−2+ζ412$ $ζ41−19+ζ4119$ $ζ41−5+ζ415$ $ζ41−12+ζ4112$ 82.1.2a5 R $2$ $0$ $ζ41−16+ζ4116$ $ζ41−9+ζ419$ $ζ41−7+ζ417$ $ζ41−18+ζ4118$ $ζ41−2+ζ412$ $ζ41−14+ζ4114$ $ζ41−11+ζ4111$ $ζ41−5+ζ415$ $ζ41−20+ζ4120$ $ζ41−4+ζ414$ $ζ41−12+ζ4112$ $ζ41−13+ζ4113$ $ζ41−3+ζ413$ $ζ41−19+ζ4119$ $ζ41−6+ζ416$ $ζ41−10+ζ4110$ $ζ41−15+ζ4115$ $ζ41−1+ζ41$ $ζ41−17+ζ4117$ $ζ41−8+ζ418$ 82.1.2a6 R $2$ $0$ $ζ41−15+ζ4115$ $ζ41−11+ζ4111$ $ζ41−4+ζ414$ $ζ41−19+ζ4119$ $ζ41−7+ζ417$ $ζ41−8+ζ418$ $ζ41−18+ζ4118$ $ζ41−3+ζ413$ $ζ41−12+ζ4112$ $ζ41−14+ζ4114$ $ζ41−1+ζ41$ $ζ41−16+ζ4116$ $ζ41−10+ζ4110$ $ζ41−5+ζ415$ $ζ41−20+ζ4120$ $ζ41−6+ζ416$ $ζ41−9+ζ419$ $ζ41−17+ζ4117$ $ζ41−2+ζ412$ $ζ41−13+ζ4113$ 82.1.2a7 R $2$ $0$ $ζ41−14+ζ4114$ $ζ41−13+ζ4113$ $ζ41−1+ζ41$ $ζ41−15+ζ4115$ $ζ41−12+ζ4112$ $ζ41−2+ζ412$ $ζ41−16+ζ4116$ $ζ41−11+ζ4111$ $ζ41−3+ζ413$ $ζ41−17+ζ4117$ $ζ41−10+ζ4110$ $ζ41−4+ζ414$ $ζ41−18+ζ4118$ $ζ41−9+ζ419$ $ζ41−5+ζ415$ $ζ41−19+ζ4119$ $ζ41−8+ζ418$ $ζ41−6+ζ416$ $ζ41−20+ζ4120$ $ζ41−7+ζ417$ 82.1.2a8 R $2$ $0$ $ζ41−13+ζ4113$ $ζ41−15+ζ4115$ $ζ41−2+ζ412$ $ζ41−11+ζ4111$ $ζ41−17+ζ4117$ $ζ41−4+ζ414$ $ζ41−9+ζ419$ $ζ41−19+ζ4119$ $ζ41−6+ζ416$ $ζ41−7+ζ417$ $ζ41−20+ζ4120$ $ζ41−8+ζ418$ $ζ41−5+ζ415$ $ζ41−18+ζ4118$ $ζ41−10+ζ4110$ $ζ41−3+ζ413$ $ζ41−16+ζ4116$ $ζ41−12+ζ4112$ $ζ41−1+ζ41$ $ζ41−14+ζ4114$ 82.1.2a9 R $2$ $0$ $ζ41−12+ζ4112$ $ζ41−17+ζ4117$ $ζ41−5+ζ415$ $ζ41−7+ζ417$ $ζ41−19+ζ4119$ $ζ41−10+ζ4110$ $ζ41−2+ζ412$ $ζ41−14+ζ4114$ $ζ41−15+ζ4115$ $ζ41−3+ζ413$ $ζ41−9+ζ419$ $ζ41−20+ζ4120$ $ζ41−8+ζ418$ $ζ41−4+ζ414$ $ζ41−16+ζ4116$ $ζ41−13+ζ4113$ $ζ41−1+ζ41$ $ζ41−11+ζ4111$ $ζ41−18+ζ4118$ $ζ41−6+ζ416$ 82.1.2a10 R $2$ $0$ $ζ41−11+ζ4111$ $ζ41−19+ζ4119$ $ζ41−8+ζ418$ $ζ41−3+ζ413$ $ζ41−14+ζ4114$ $ζ41−16+ζ4116$ $ζ41−5+ζ415$ $ζ41−6+ζ416$ $ζ41−17+ζ4117$ $ζ41−13+ζ4113$ $ζ41−2+ζ412$ $ζ41−9+ζ419$ $ζ41−20+ζ4120$ $ζ41−10+ζ4110$ $ζ41−1+ζ41$ $ζ41−12+ζ4112$ $ζ41−18+ζ4118$ $ζ41−7+ζ417$ $ζ41−4+ζ414$ $ζ41−15+ζ4115$ 82.1.2a11 R $2$ $0$ $ζ41−10+ζ4110$ $ζ41−20+ζ4120$ $ζ41−11+ζ4111$ $ζ41−1+ζ41$ $ζ41−9+ζ419$ $ζ41−19+ζ4119$ $ζ41−12+ζ4112$ $ζ41−2+ζ412$ $ζ41−8+ζ418$ $ζ41−18+ζ4118$ $ζ41−13+ζ4113$ $ζ41−3+ζ413$ $ζ41−7+ζ417$ $ζ41−17+ζ4117$ $ζ41−14+ζ4114$ $ζ41−4+ζ414$ $ζ41−6+ζ416$ $ζ41−16+ζ4116$ $ζ41−15+ζ4115$ $ζ41−5+ζ415$ 82.1.2a12 R $2$ $0$ $ζ41−9+ζ419$ $ζ41−18+ζ4118$ $ζ41−14+ζ4114$ $ζ41−5+ζ415$ $ζ41−4+ζ414$ $ζ41−13+ζ4113$ $ζ41−19+ζ4119$ $ζ41−10+ζ4110$ $ζ41−1+ζ41$ $ζ41−8+ζ418$ $ζ41−17+ζ4117$ $ζ41−15+ζ4115$ $ζ41−6+ζ416$ $ζ41−3+ζ413$ $ζ41−12+ζ4112$ $ζ41−20+ζ4120$ $ζ41−11+ζ4111$ $ζ41−2+ζ412$ $ζ41−7+ζ417$ $ζ41−16+ζ4116$ 82.1.2a13 R $2$ $0$ $ζ41−8+ζ418$ $ζ41−16+ζ4116$ $ζ41−17+ζ4117$ $ζ41−9+ζ419$ $ζ41−1+ζ41$ $ζ41−7+ζ417$ $ζ41−15+ζ4115$ $ζ41−18+ζ4118$ $ζ41−10+ζ4110$ $ζ41−2+ζ412$ $ζ41−6+ζ416$ $ζ41−14+ζ4114$ $ζ41−19+ζ4119$ $ζ41−11+ζ4111$ $ζ41−3+ζ413$ $ζ41−5+ζ415$ $ζ41−13+ζ4113$ $ζ41−20+ζ4120$ $ζ41−12+ζ4112$ $ζ41−4+ζ414$ 82.1.2a14 R $2$ $0$ $ζ41−7+ζ417$ $ζ41−14+ζ4114$ $ζ41−20+ζ4120$ $ζ41−13+ζ4113$ $ζ41−6+ζ416$ $ζ41−1+ζ41$ $ζ41−8+ζ418$ $ζ41−15+ζ4115$ $ζ41−19+ζ4119$ $ζ41−12+ζ4112$ $ζ41−5+ζ415$ $ζ41−2+ζ412$ $ζ41−9+ζ419$ $ζ41−16+ζ4116$ $ζ41−18+ζ4118$ $ζ41−11+ζ4111$ $ζ41−4+ζ414$ $ζ41−3+ζ413$ $ζ41−10+ζ4110$ $ζ41−17+ζ4117$ 82.1.2a15 R $2$ $0$ $ζ41−6+ζ416$ $ζ41−12+ζ4112$ $ζ41−18+ζ4118$ $ζ41−17+ζ4117$ $ζ41−11+ζ4111$ $ζ41−5+ζ415$ $ζ41−1+ζ41$ $ζ41−7+ζ417$ $ζ41−13+ζ4113$ $ζ41−19+ζ4119$ $ζ41−16+ζ4116$ $ζ41−10+ζ4110$ $ζ41−4+ζ414$ $ζ41−2+ζ412$ $ζ41−8+ζ418$ $ζ41−14+ζ4114$ $ζ41−20+ζ4120$ $ζ41−15+ζ4115$ $ζ41−9+ζ419$ $ζ41−3+ζ413$ 82.1.2a16 R $2$ $0$ $ζ41−5+ζ415$ $ζ41−10+ζ4110$ $ζ41−15+ζ4115$ $ζ41−20+ζ4120$ $ζ41−16+ζ4116$ $ζ41−11+ζ4111$ $ζ41−6+ζ416$ $ζ41−1+ζ41$ $ζ41−4+ζ414$ $ζ41−9+ζ419$ $ζ41−14+ζ4114$ $ζ41−19+ζ4119$ $ζ41−17+ζ4117$ $ζ41−12+ζ4112$ $ζ41−7+ζ417$ $ζ41−2+ζ412$ $ζ41−3+ζ413$ $ζ41−8+ζ418$ $ζ41−13+ζ4113$ $ζ41−18+ζ4118$ 82.1.2a17 R $2$ $0$ $ζ41−4+ζ414$ $ζ41−8+ζ418$ $ζ41−12+ζ4112$ $ζ41−16+ζ4116$ $ζ41−20+ζ4120$ $ζ41−17+ζ4117$ $ζ41−13+ζ4113$ $ζ41−9+ζ419$ $ζ41−5+ζ415$ $ζ41−1+ζ41$ $ζ41−3+ζ413$ $ζ41−7+ζ417$ $ζ41−11+ζ4111$ $ζ41−15+ζ4115$ $ζ41−19+ζ4119$ $ζ41−18+ζ4118$ $ζ41−14+ζ4114$ $ζ41−10+ζ4110$ $ζ41−6+ζ416$ $ζ41−2+ζ412$ 82.1.2a18 R $2$ $0$ $ζ41−3+ζ413$ $ζ41−6+ζ416$ $ζ41−9+ζ419$ $ζ41−12+ζ4112$ $ζ41−15+ζ4115$ $ζ41−18+ζ4118$ $ζ41−20+ζ4120$ $ζ41−17+ζ4117$ $ζ41−14+ζ4114$ $ζ41−11+ζ4111$ $ζ41−8+ζ418$ $ζ41−5+ζ415$ $ζ41−2+ζ412$ $ζ41−1+ζ41$ $ζ41−4+ζ414$ $ζ41−7+ζ417$ $ζ41−10+ζ4110$ $ζ41−13+ζ4113$ $ζ41−16+ζ4116$ $ζ41−19+ζ4119$ 82.1.2a19 R $2$ $0$ $ζ41−2+ζ412$ $ζ41−4+ζ414$ $ζ41−6+ζ416$ $ζ41−8+ζ418$ $ζ41−10+ζ4110$ $ζ41−12+ζ4112$ $ζ41−14+ζ4114$ $ζ41−16+ζ4116$ $ζ41−18+ζ4118$ $ζ41−20+ζ4120$ $ζ41−19+ζ4119$ $ζ41−17+ζ4117$ $ζ41−15+ζ4115$ $ζ41−13+ζ4113$ $ζ41−11+ζ4111$ $ζ41−9+ζ419$ $ζ41−7+ζ417$ $ζ41−5+ζ415$ $ζ41−3+ζ413$ $ζ41−1+ζ41$ 82.1.2a20 R $2$ $0$ $ζ41−1+ζ41$ $ζ41−2+ζ412$ $ζ41−3+ζ413$ $ζ41−4+ζ414$ $ζ41−5+ζ415$ $ζ41−6+ζ416$ $ζ41−7+ζ417$ $ζ41−8+ζ418$ $ζ41−9+ζ419$ $ζ41−10+ζ4110$ $ζ41−11+ζ4111$ $ζ41−12+ζ4112$ $ζ41−13+ζ4113$ $ζ41−14+ζ4114$ $ζ41−15+ζ4115$ $ζ41−16+ζ4116$ $ζ41−17+ζ4117$ $ζ41−18+ζ4118$ $ζ41−19+ζ4119$ $ζ41−20+ζ4120$

magma: CharacterTable(G);