Properties

Label 41T4
Degree $41$
Order $205$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{41}:C_{5}$

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Show commands: Magma

magma: G := TransitiveGroup(41, 4);
 

Group action invariants

Degree $n$:  $41$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{41}:C_{5}$
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:  (1,10,18,16,37)(2,20,36,32,33)(3,30,13,7,29)(4,40,31,23,25)(5,9,8,39,21)(6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22), (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)
$5$:  $C_5$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $205=5 \cdot 41$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  205.1
magma: IdentifyGroup(G);
 
Character table:

1A 5A1 5A-1 5A2 5A-2 41A1 41A-1 41A2 41A-2 41A3 41A-3 41A6 41A-6
Size 1 41 41 41 41 5 5 5 5 5 5 5 5
5 P 1A 5A-1 5A1 5A2 5A-2 41A3 41A6 41A-1 41A-2 41A1 41A-6 41A2 41A-3
41 P 1A 1A 1A 1A 1A 41A-3 41A-6 41A1 41A2 41A-1 41A6 41A-2 41A3
Type
205.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
205.1.1b1 C 1 ζ52 ζ52 ζ5 ζ51 1 1 1 1 1 1 1 1
205.1.1b2 C 1 ζ52 ζ52 ζ51 ζ5 1 1 1 1 1 1 1 1
205.1.1b3 C 1 ζ51 ζ5 ζ52 ζ52 1 1 1 1 1 1 1 1
205.1.1b4 C 1 ζ5 ζ51 ζ52 ζ52 1 1 1 1 1 1 1 1
205.1.5a1 C 5 0 0 0 0 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ419+ζ418+ζ415+ζ412+ζ4120
205.1.5a2 C 5 0 0 0 0 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ4120+ζ412+ζ415+ζ418+ζ419
205.1.5a3 C 5 0 0 0 0 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4115+ζ416+ζ4114+ζ4117+ζ4119
205.1.5a4 C 5 0 0 0 0 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4119+ζ4117+ζ4114+ζ416+ζ4115
205.1.5a5 C 5 0 0 0 0 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ414+ζ41+ζ4110+ζ4116+ζ4118
205.1.5a6 C 5 0 0 0 0 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4118+ζ4116+ζ4110+ζ411+ζ414
205.1.5a7 C 5 0 0 0 0 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4113+ζ417+ζ413+ζ4111+ζ4112 ζ4112+ζ4111+ζ413+ζ417+ζ4113
205.1.5a8 C 5 0 0 0 0 ζ4120+ζ412+ζ415+ζ418+ζ419 ζ419+ζ418+ζ415+ζ412+ζ4120 ζ414+ζ41+ζ4110+ζ4116+ζ4118 ζ4118+ζ4116+ζ4110+ζ411+ζ414 ζ4119+ζ4117+ζ4114+ζ416+ζ4115 ζ4115+ζ416+ζ4114+ζ4117+ζ4119 ζ4112+ζ4111+ζ413+ζ417+ζ4113 ζ4113+ζ417+ζ413+ζ4111+ζ4112

magma: CharacterTable(G);