Properties

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

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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

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

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 ζ4120+ζ4120 ζ411+ζ41 ζ4119+ζ4119 ζ412+ζ412 ζ4118+ζ4118 ζ413+ζ413 ζ4117+ζ4117 ζ414+ζ414 ζ4116+ζ4116 ζ415+ζ415 ζ4115+ζ4115 ζ416+ζ416 ζ4114+ζ4114 ζ417+ζ417 ζ4113+ζ4113 ζ418+ζ418 ζ4112+ζ4112 ζ419+ζ419 ζ4111+ζ4111 ζ4110+ζ4110
82.1.2a2 R 2 0 ζ4119+ζ4119 ζ413+ζ413 ζ4116+ζ4116 ζ416+ζ416 ζ4113+ζ4113 ζ419+ζ419 ζ4110+ζ4110 ζ4112+ζ4112 ζ417+ζ417 ζ4115+ζ4115 ζ414+ζ414 ζ4118+ζ4118 ζ411+ζ41 ζ4120+ζ4120 ζ412+ζ412 ζ4117+ζ4117 ζ415+ζ415 ζ4114+ζ4114 ζ418+ζ418 ζ4111+ζ4111
82.1.2a3 R 2 0 ζ4118+ζ4118 ζ415+ζ415 ζ4113+ζ4113 ζ4110+ζ4110 ζ418+ζ418 ζ4115+ζ4115 ζ413+ζ413 ζ4120+ζ4120 ζ412+ζ412 ζ4116+ζ4116 ζ417+ζ417 ζ4111+ζ4111 ζ4112+ζ4112 ζ416+ζ416 ζ4117+ζ4117 ζ411+ζ41 ζ4119+ζ4119 ζ414+ζ414 ζ4114+ζ4114 ζ419+ζ419
82.1.2a4 R 2 0 ζ4117+ζ4117 ζ417+ζ417 ζ4110+ζ4110 ζ4114+ζ4114 ζ413+ζ413 ζ4120+ζ4120 ζ414+ζ414 ζ4113+ζ4113 ζ4111+ζ4111 ζ416+ζ416 ζ4118+ζ4118 ζ411+ζ41 ζ4116+ζ4116 ζ418+ζ418 ζ419+ζ419 ζ4115+ζ4115 ζ412+ζ412 ζ4119+ζ4119 ζ415+ζ415 ζ4112+ζ4112
82.1.2a5 R 2 0 ζ4116+ζ4116 ζ419+ζ419 ζ417+ζ417 ζ4118+ζ4118 ζ412+ζ412 ζ4114+ζ4114 ζ4111+ζ4111 ζ415+ζ415 ζ4120+ζ4120 ζ414+ζ414 ζ4112+ζ4112 ζ4113+ζ4113 ζ413+ζ413 ζ4119+ζ4119 ζ416+ζ416 ζ4110+ζ4110 ζ4115+ζ4115 ζ411+ζ41 ζ4117+ζ4117 ζ418+ζ418
82.1.2a6 R 2 0 ζ4115+ζ4115 ζ4111+ζ4111 ζ414+ζ414 ζ4119+ζ4119 ζ417+ζ417 ζ418+ζ418 ζ4118+ζ4118 ζ413+ζ413 ζ4112+ζ4112 ζ4114+ζ4114 ζ411+ζ41 ζ4116+ζ4116 ζ4110+ζ4110 ζ415+ζ415 ζ4120+ζ4120 ζ416+ζ416 ζ419+ζ419 ζ4117+ζ4117 ζ412+ζ412 ζ4113+ζ4113
82.1.2a7 R 2 0 ζ4114+ζ4114 ζ4113+ζ4113 ζ411+ζ41 ζ4115+ζ4115 ζ4112+ζ4112 ζ412+ζ412 ζ4116+ζ4116 ζ4111+ζ4111 ζ413+ζ413 ζ4117+ζ4117 ζ4110+ζ4110 ζ414+ζ414 ζ4118+ζ4118 ζ419+ζ419 ζ415+ζ415 ζ4119+ζ4119 ζ418+ζ418 ζ416+ζ416 ζ4120+ζ4120 ζ417+ζ417
82.1.2a8 R 2 0 ζ4113+ζ4113 ζ4115+ζ4115 ζ412+ζ412 ζ4111+ζ4111 ζ4117+ζ4117 ζ414+ζ414 ζ419+ζ419 ζ4119+ζ4119 ζ416+ζ416 ζ417+ζ417 ζ4120+ζ4120 ζ418+ζ418 ζ415+ζ415 ζ4118+ζ4118 ζ4110+ζ4110 ζ413+ζ413 ζ4116+ζ4116 ζ4112+ζ4112 ζ411+ζ41 ζ4114+ζ4114
82.1.2a9 R 2 0 ζ4112+ζ4112 ζ4117+ζ4117 ζ415+ζ415 ζ417+ζ417 ζ4119+ζ4119 ζ4110+ζ4110 ζ412+ζ412 ζ4114+ζ4114 ζ4115+ζ4115 ζ413+ζ413 ζ419+ζ419 ζ4120+ζ4120 ζ418+ζ418 ζ414+ζ414 ζ4116+ζ4116 ζ4113+ζ4113 ζ411+ζ41 ζ4111+ζ4111 ζ4118+ζ4118 ζ416+ζ416
82.1.2a10 R 2 0 ζ4111+ζ4111 ζ4119+ζ4119 ζ418+ζ418 ζ413+ζ413 ζ4114+ζ4114 ζ4116+ζ4116 ζ415+ζ415 ζ416+ζ416 ζ4117+ζ4117 ζ4113+ζ4113 ζ412+ζ412 ζ419+ζ419 ζ4120+ζ4120 ζ4110+ζ4110 ζ411+ζ41 ζ4112+ζ4112 ζ4118+ζ4118 ζ417+ζ417 ζ414+ζ414 ζ4115+ζ4115
82.1.2a11 R 2 0 ζ4110+ζ4110 ζ4120+ζ4120 ζ4111+ζ4111 ζ411+ζ41 ζ419+ζ419 ζ4119+ζ4119 ζ4112+ζ4112 ζ412+ζ412 ζ418+ζ418 ζ4118+ζ4118 ζ4113+ζ4113 ζ413+ζ413 ζ417+ζ417 ζ4117+ζ4117 ζ4114+ζ4114 ζ414+ζ414 ζ416+ζ416 ζ4116+ζ4116 ζ4115+ζ4115 ζ415+ζ415
82.1.2a12 R 2 0 ζ419+ζ419 ζ4118+ζ4118 ζ4114+ζ4114 ζ415+ζ415 ζ414+ζ414 ζ4113+ζ4113 ζ4119+ζ4119 ζ4110+ζ4110 ζ411+ζ41 ζ418+ζ418 ζ4117+ζ4117 ζ4115+ζ4115 ζ416+ζ416 ζ413+ζ413 ζ4112+ζ4112 ζ4120+ζ4120 ζ4111+ζ4111 ζ412+ζ412 ζ417+ζ417 ζ4116+ζ4116
82.1.2a13 R 2 0 ζ418+ζ418 ζ4116+ζ4116 ζ4117+ζ4117 ζ419+ζ419 ζ411+ζ41 ζ417+ζ417 ζ4115+ζ4115 ζ4118+ζ4118 ζ4110+ζ4110 ζ412+ζ412 ζ416+ζ416 ζ4114+ζ4114 ζ4119+ζ4119 ζ4111+ζ4111 ζ413+ζ413 ζ415+ζ415 ζ4113+ζ4113 ζ4120+ζ4120 ζ4112+ζ4112 ζ414+ζ414
82.1.2a14 R 2 0 ζ417+ζ417 ζ4114+ζ4114 ζ4120+ζ4120 ζ4113+ζ4113 ζ416+ζ416 ζ411+ζ41 ζ418+ζ418 ζ4115+ζ4115 ζ4119+ζ4119 ζ4112+ζ4112 ζ415+ζ415 ζ412+ζ412 ζ419+ζ419 ζ4116+ζ4116 ζ4118+ζ4118 ζ4111+ζ4111 ζ414+ζ414 ζ413+ζ413 ζ4110+ζ4110 ζ4117+ζ4117
82.1.2a15 R 2 0 ζ416+ζ416 ζ4112+ζ4112 ζ4118+ζ4118 ζ4117+ζ4117 ζ4111+ζ4111 ζ415+ζ415 ζ411+ζ41 ζ417+ζ417 ζ4113+ζ4113 ζ4119+ζ4119 ζ4116+ζ4116 ζ4110+ζ4110 ζ414+ζ414 ζ412+ζ412 ζ418+ζ418 ζ4114+ζ4114 ζ4120+ζ4120 ζ4115+ζ4115 ζ419+ζ419 ζ413+ζ413
82.1.2a16 R 2 0 ζ415+ζ415 ζ4110+ζ4110 ζ4115+ζ4115 ζ4120+ζ4120 ζ4116+ζ4116 ζ4111+ζ4111 ζ416+ζ416 ζ411+ζ41 ζ414+ζ414 ζ419+ζ419 ζ4114+ζ4114 ζ4119+ζ4119 ζ4117+ζ4117 ζ4112+ζ4112 ζ417+ζ417 ζ412+ζ412 ζ413+ζ413 ζ418+ζ418 ζ4113+ζ4113 ζ4118+ζ4118
82.1.2a17 R 2 0 ζ414+ζ414 ζ418+ζ418 ζ4112+ζ4112 ζ4116+ζ4116 ζ4120+ζ4120 ζ4117+ζ4117 ζ4113+ζ4113 ζ419+ζ419 ζ415+ζ415 ζ411+ζ41 ζ413+ζ413 ζ417+ζ417 ζ4111+ζ4111 ζ4115+ζ4115 ζ4119+ζ4119 ζ4118+ζ4118 ζ4114+ζ4114 ζ4110+ζ4110 ζ416+ζ416 ζ412+ζ412
82.1.2a18 R 2 0 ζ413+ζ413 ζ416+ζ416 ζ419+ζ419 ζ4112+ζ4112 ζ4115+ζ4115 ζ4118+ζ4118 ζ4120+ζ4120 ζ4117+ζ4117 ζ4114+ζ4114 ζ4111+ζ4111 ζ418+ζ418 ζ415+ζ415 ζ412+ζ412 ζ411+ζ41 ζ414+ζ414 ζ417+ζ417 ζ4110+ζ4110 ζ4113+ζ4113 ζ4116+ζ4116 ζ4119+ζ4119
82.1.2a19 R 2 0 ζ412+ζ412 ζ414+ζ414 ζ416+ζ416 ζ418+ζ418 ζ4110+ζ4110 ζ4112+ζ4112 ζ4114+ζ4114 ζ4116+ζ4116 ζ4118+ζ4118 ζ4120+ζ4120 ζ4119+ζ4119 ζ4117+ζ4117 ζ4115+ζ4115 ζ4113+ζ4113 ζ4111+ζ4111 ζ419+ζ419 ζ417+ζ417 ζ415+ζ415 ζ413+ζ413 ζ411+ζ41
82.1.2a20 R 2 0 ζ411+ζ41 ζ412+ζ412 ζ413+ζ413 ζ414+ζ414 ζ415+ζ415 ζ416+ζ416 ζ417+ζ417 ζ418+ζ418 ζ419+ζ419 ζ4110+ζ4110 ζ4111+ζ4111 ζ4112+ζ4112 ζ4113+ζ4113 ζ4114+ζ4114 ζ4115+ζ4115 ζ4116+ζ4116 ζ4117+ζ4117 ζ4118+ζ4118 ζ4119+ζ4119 ζ4120+ζ4120

magma: CharacterTable(G);