# Properties

 Label 41T6 Degree $41$ Order $410$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{41}:C_{10}$

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magma: G := TransitiveGroup(41, 6);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $410=2 \cdot 5 \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 410.1 magma: IdentifyGroup(G); Character table:

 1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 41A1 41A2 41A3 41A6 Size 1 41 41 41 41 41 41 41 41 41 10 10 10 10 2 P 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 41A1 41A3 41A6 41A2 5 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 41A1 41A3 41A6 41A2 41 P 1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 1A 1A 1A 1A Type 410.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 410.1.1b R $1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ $1$ 410.1.1c1 C $1$ $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $1$ $1$ $1$ $1$ 410.1.1c2 C $1$ $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $1$ $1$ $1$ $1$ 410.1.1c3 C $1$ $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $1$ $1$ $1$ $1$ 410.1.1c4 C $1$ $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $1$ $1$ $1$ $1$ 410.1.1d1 C $1$ $−1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $1$ $1$ $1$ $1$ 410.1.1d2 C $1$ $−1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $1$ $1$ $1$ $1$ 410.1.1d3 C $1$ $−1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $1$ $1$ $1$ $1$ 410.1.1d4 C $1$ $−1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $1$ $1$ $1$ $1$ 410.1.10a1 R $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−6+ζ416+ζ4114+ζ4115+ζ4117+ζ4119$ $ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−3+ζ413+ζ417+ζ4111+ζ4112+ζ4113$ $ζ41−18+ζ41−16+ζ41−10+ζ41−4+ζ41−1+ζ41+ζ414+ζ4110+ζ4116+ζ4118$ $ζ41−20+ζ41−9+ζ41−8+ζ41−5+ζ41−2+ζ412+ζ415+ζ418+ζ419+ζ4120$ 410.1.10a2 R $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−18+ζ41−16+ζ41−10+ζ41−4+ζ41−1+ζ41+ζ414+ζ4110+ζ4116+ζ4118$ $ζ41−20+ζ41−9+ζ41−8+ζ41−5+ζ41−2+ζ412+ζ415+ζ418+ζ419+ζ4120$ $ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−3+ζ413+ζ417+ζ4111+ζ4112+ζ4113$ $ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−6+ζ416+ζ4114+ζ4115+ζ4117+ζ4119$ 410.1.10a3 R $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−3+ζ413+ζ417+ζ4111+ζ4112+ζ4113$ $ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−6+ζ416+ζ4114+ζ4115+ζ4117+ζ4119$ $ζ41−20+ζ41−9+ζ41−8+ζ41−5+ζ41−2+ζ412+ζ415+ζ418+ζ419+ζ4120$ $ζ41−18+ζ41−16+ζ41−10+ζ41−4+ζ41−1+ζ41+ζ414+ζ4110+ζ4116+ζ4118$ 410.1.10a4 R $10$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−20+ζ41−9+ζ41−8+ζ41−5+ζ41−2+ζ412+ζ415+ζ418+ζ419+ζ4120$ $ζ41−18+ζ41−16+ζ41−10+ζ41−4+ζ41−1+ζ41+ζ414+ζ4110+ζ4116+ζ4118$ $ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−6+ζ416+ζ4114+ζ4115+ζ4117+ζ4119$ $ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−3+ζ413+ζ417+ζ4111+ζ4112+ζ4113$

magma: CharacterTable(G);