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Properties

Label 41T7
Degree $41$
Order $820$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{41}:C_{20}$

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Show commands: Magma

magma: G := TransitiveGroup(41, 7);
 

Group action invariants

Degree $n$:  $41$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{41}:C_{20}$
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:  (1,36,25,39,10,32,4,21,18,33,40,5,16,2,31,9,37,20,23,8)(3,26,34,35,30,14,12,22,13,17,38,15,7,6,11,27,29,19,28,24), (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)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  20T1

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 TypeSizeOrder IndexRepresentative
1A $1^{41}$ $1$ $1$ $0$ $()$
2A $2^{20},1$ $41$ $2$ $20$ $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$
4A1 $4^{10},1$ $41$ $4$ $30$ $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20)( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$
4A-1 $4^{10},1$ $41$ $4$ $30$ $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23)( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$
5A1 $5^{8},1$ $41$ $5$ $32$ $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32)( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$
5A-1 $5^{8},1$ $41$ $5$ $32$ $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24)( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$
5A2 $5^{8},1$ $41$ $5$ $32$ $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26)( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$
5A-2 $5^{8},1$ $41$ $5$ $32$ $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41)( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$
10A1 $10^{4},1$ $41$ $10$ $36$ $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31,39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$
10A-1 $10^{4},1$ $41$ $10$ $36$ $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30,39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$
10A3 $10^{4},1$ $41$ $10$ $36$ $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8,39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$
10A-3 $10^{4},1$ $41$ $10$ $36$ $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14,39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$
20A1 $20^{2},1$ $41$ $20$ $38$ $( 2, 3, 5, 9,17,33,24, 6,11,21,41,40,38,34,26,10,19,37,32,22)( 4, 7,13,25, 8,15,29,16,31,20,39,36,30,18,35,28,14,27,12,23)$
20A-1 $20^{2},1$ $41$ $20$ $38$ $( 2,40, 5,34,17,10,24,37,11,22,41, 3,38, 9,26,33,19, 6,32,21)( 4,36,13,18, 8,28,29,27,31,23,39, 7,30,25,35,15,14,16,12,20)$
20A3 $20^{2},1$ $41$ $20$ $38$ $( 2,34,24,22,38,33,32,40,17,37,41, 9,19,21, 5,10,11, 3,26, 6)( 4,18,29,23,30,15,12,36, 8,27,39,25,14,20,13,28,31, 7,35,16)$
20A-3 $20^{2},1$ $41$ $20$ $38$ $( 2, 9,24,21,38,10,32, 3,17, 6,41,34,19,22, 5,33,11,40,26,37)( 4,25,29,20,30,28,12, 7, 8,16,39,18,14,23,13,15,31,36,35,27)$
20A7 $20^{2},1$ $41$ $20$ $38$ $( 2, 6,26, 3,11,10, 5,21,19, 9,41,37,17,40,32,33,38,22,24,34)( 4,16,35, 7,31,28,13,20,14,25,39,27, 8,36,12,15,30,23,29,18)$
20A-7 $20^{2},1$ $41$ $20$ $38$ $( 2,37,26,40,11,33, 5,22,19,34,41, 6,17, 3,32,10,38,21,24, 9)( 4,27,35,36,31,15,13,23,14,18,39,16, 8, 7,12,28,30,20,29,25)$
20A9 $20^{2},1$ $41$ $20$ $38$ $( 2,22,32,37,19,10,26,34,38,40,41,21,11, 6,24,33,17, 9, 5, 3)( 4,23,12,27,14,28,35,18,30,36,39,20,31,16,29,15, 8,25,13, 7)$
20A-9 $20^{2},1$ $41$ $20$ $38$ $( 2,21,32, 6,19,33,26, 9,38, 3,41,22,11,37,24,10,17,34, 5,40)( 4,20,12,16,14,15,35,25,30, 7,39,23,31,27,29,28, 8,18,13,36)$
41A1 $41$ $20$ $41$ $40$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$
41A3 $41$ $20$ $41$ $40$ $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/20$

Group invariants

Order:  $820=2^{2} \cdot 5 \cdot 41$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  820.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9 41A1 41A3
Size 1 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 20 20
2 P 1A 1A 2A 2A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 10A1 10A1 10A3 10A3 10A-3 10A-3 10A-1 10A-1 41A1 41A3
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 2A 2A 2A 2A 4A-1 4A1 4A-1 4A1 4A1 4A-1 4A1 4A-1 41A1 41A3
41 P 1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A-9 20A1 20A3 20A-7 20A-3 20A7 20A9 20A-1 1A 1A
Type
820.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
820.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
820.7.1c1 C 1 −1 −i i 1 1 1 1 −1 −1 −1 −1 −i i i −i i −i −i i 1 1
820.7.1c2 C 1 −1 i −i 1 1 1 1 −1 −1 −1 −1 i −i −i i −i i i −i 1 1
820.7.1d1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 1 1
820.7.1d2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 1 1
820.7.1d3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 1 1
820.7.1d4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 1 1
820.7.1e1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52 1 1
820.7.1e2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2 1 1
820.7.1e3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5 1 1
820.7.1e4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1 1 1
820.7.1f1 C 1 −1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 ζ203 −ζ207 ζ209 −ζ20 ζ20 −ζ209 ζ207 −ζ203 1 1
820.7.1f2 C 1 −1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 −ζ207 ζ203 −ζ20 ζ209 −ζ209 ζ20 −ζ203 ζ207 1 1
820.7.1f3 C 1 −1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 −ζ204 ζ206 ζ202 −ζ208 ζ207 −ζ203 ζ20 −ζ209 ζ209 −ζ20 ζ203 −ζ207 1 1
820.7.1f4 C 1 −1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 ζ206 −ζ204 −ζ208 ζ202 −ζ203 ζ207 −ζ209 ζ20 −ζ20 ζ209 −ζ207 ζ203 1 1
820.7.1f5 C 1 −1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 −ζ209 ζ20 −ζ207 ζ203 −ζ203 ζ207 −ζ20 ζ209 1 1
820.7.1f6 C 1 −1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 ζ20 −ζ209 ζ203 −ζ207 ζ207 −ζ203 ζ209 −ζ20 1 1
820.7.1f7 C 1 −1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ202 −ζ208 ζ206 −ζ204 −ζ20 ζ209 −ζ203 ζ207 −ζ207 ζ203 −ζ209 ζ20 1 1
820.7.1f8 C 1 −1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ208 ζ202 −ζ204 ζ206 ζ209 −ζ20 ζ207 −ζ203 ζ203 −ζ207 ζ20 −ζ209 1 1
820.7.20a1 R 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−6+ζ41−3+ζ413+ζ416+ζ417+ζ4111+ζ4112+ζ4113+ζ4114+ζ4115+ζ4117+ζ4119 −ζ41−19−ζ41−17−ζ41−15−ζ41−14−ζ41−13−ζ41−12−ζ41−11−ζ41−7−ζ41−6−ζ41−3−1−ζ413−ζ416−ζ417−ζ4111−ζ4112−ζ4113−ζ4114−ζ4115−ζ4117−ζ4119
820.7.20a2 R 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ41−19−ζ41−17−ζ41−15−ζ41−14−ζ41−13−ζ41−12−ζ41−11−ζ41−7−ζ41−6−ζ41−3−1−ζ413−ζ416−ζ417−ζ4111−ζ4112−ζ4113−ζ4114−ζ4115−ζ4117−ζ4119 ζ41−19+ζ41−17+ζ41−15+ζ41−14+ζ41−13+ζ41−12+ζ41−11+ζ41−7+ζ41−6+ζ41−3+ζ413+ζ416+ζ417+ζ4111+ζ4112+ζ4113+ζ4114+ζ4115+ζ4117+ζ4119

magma: CharacterTable(G);