Properties

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

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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

LabelCycle TypeSizeOrderRepresentative
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 ζ4120+ζ4116+ζ4116+ζ4120 ζ419+ζ411+ζ41+ζ419 ζ4119+ζ417+ζ417+ζ4119 ζ4118+ζ412+ζ412+ζ4118 ζ4114+ζ413+ζ413+ζ4114 ζ4117+ζ4111+ζ4111+ζ4117 ζ415+ζ414+ζ414+ζ415 ζ4115+ζ4112+ζ4112+ζ4115 ζ4113+ζ416+ζ416+ζ4113 ζ4110+ζ418+ζ418+ζ4110
164.3.4a2 R 4 0 0 0 ζ4117+ζ4111+ζ4111+ζ4117 ζ4119+ζ417+ζ417+ζ4119 ζ4110+ζ418+ζ418+ζ4110 ζ4114+ζ413+ζ413+ζ4114 ζ4120+ζ4116+ζ4116+ζ4120 ζ415+ζ414+ζ414+ζ415 ζ4113+ζ416+ζ416+ζ4113 ζ4118+ζ412+ζ412+ζ4118 ζ419+ζ411+ζ41+ζ419 ζ4115+ζ4112+ζ4112+ζ4115
164.3.4a3 R 4 0 0 0 ζ4115+ζ4112+ζ4112+ζ4115 ζ4117+ζ4111+ζ4111+ζ4117 ζ415+ζ414+ζ414+ζ415 ζ4119+ζ417+ζ417+ζ4119 ζ4110+ζ418+ζ418+ζ4110 ζ4118+ζ412+ζ412+ζ4118 ζ4114+ζ413+ζ413+ζ4114 ζ419+ζ411+ζ41+ζ419 ζ4120+ζ4116+ζ4116+ζ4120 ζ4113+ζ416+ζ416+ζ4113
164.3.4a4 R 4 0 0 0 ζ4119+ζ417+ζ417+ζ4119 ζ4114+ζ413+ζ413+ζ4114 ζ4120+ζ4116+ζ4116+ζ4120 ζ4113+ζ416+ζ416+ζ4113 ζ419+ζ411+ζ41+ζ419 ζ4110+ζ418+ζ418+ζ4110 ζ4115+ζ4112+ζ4112+ζ4115 ζ415+ζ414+ζ414+ζ415 ζ4118+ζ412+ζ412+ζ4118 ζ4117+ζ4111+ζ4111+ζ4117
164.3.4a5 R 4 0 0 0 ζ4118+ζ412+ζ412+ζ4118 ζ415+ζ414+ζ414+ζ415 ζ4113+ζ416+ζ416+ζ4113 ζ4110+ζ418+ζ418+ζ4110 ζ4115+ζ4112+ζ4112+ζ4115 ζ4114+ζ413+ζ413+ζ4114 ζ4120+ζ4116+ζ4116+ζ4120 ζ4119+ζ417+ζ417+ζ4119 ζ4117+ζ4111+ζ4111+ζ4117 ζ419+ζ411+ζ41+ζ419
164.3.4a6 R 4 0 0 0 ζ4113+ζ416+ζ416+ζ4113 ζ4115+ζ4112+ζ4112+ζ4115 ζ4118+ζ412+ζ412+ζ4118 ζ4117+ζ4111+ζ4111+ζ4117 ζ415+ζ414+ζ414+ζ415 ζ419+ζ411+ζ41+ζ419 ζ4119+ζ417+ζ417+ζ4119 ζ4120+ζ4116+ζ4116+ζ4120 ζ4110+ζ418+ζ418+ζ4110 ζ4114+ζ413+ζ413+ζ4114
164.3.4a7 R 4 0 0 0 ζ4114+ζ413+ζ413+ζ4114 ζ4113+ζ416+ζ416+ζ4113 ζ419+ζ411+ζ41+ζ419 ζ4115+ζ4112+ζ4112+ζ4115 ζ4118+ζ412+ζ412+ζ4118 ζ4120+ζ4116+ζ4116+ζ4120 ζ4117+ζ4111+ζ4111+ζ4117 ζ4110+ζ418+ζ418+ζ4110 ζ415+ζ414+ζ414+ζ415 ζ4119+ζ417+ζ417+ζ4119
164.3.4a8 R 4 0 0 0 ζ4110+ζ418+ζ418+ζ4110 ζ4120+ζ4116+ζ4116+ζ4120 ζ4117+ζ4111+ζ4111+ζ4117 ζ419+ζ411+ζ41+ζ419 ζ4119+ζ417+ζ417+ζ4119 ζ4115+ζ4112+ζ4112+ζ4115 ζ4118+ζ412+ζ412+ζ4118 ζ4113+ζ416+ζ416+ζ4113 ζ4114+ζ413+ζ413+ζ4114 ζ415+ζ414+ζ414+ζ415
164.3.4a9 R 4 0 0 0 ζ419+ζ411+ζ41+ζ419 ζ4118+ζ412+ζ412+ζ4118 ζ4114+ζ413+ζ413+ζ4114 ζ415+ζ414+ζ414+ζ415 ζ4113+ζ416+ζ416+ζ4113 ζ4119+ζ417+ζ417+ζ4119 ζ4110+ζ418+ζ418+ζ4110 ζ4117+ζ4111+ζ4111+ζ4117 ζ4115+ζ4112+ζ4112+ζ4115 ζ4120+ζ4116+ζ4116+ζ4120
164.3.4a10 R 4 0 0 0 ζ415+ζ414+ζ414+ζ415 ζ4110+ζ418+ζ418+ζ4110 ζ4115+ζ4112+ζ4112+ζ4115 ζ4120+ζ4116+ζ4116+ζ4120 ζ4117+ζ4111+ζ4111+ζ4117 ζ4113+ζ416+ζ416+ζ4113 ζ419+ζ411+ζ41+ζ419 ζ4114+ζ413+ζ413+ζ4114 ζ4119+ζ417+ζ417+ζ4119 ζ4118+ζ412+ζ412+ζ4118

magma: CharacterTable(G);