# Properties

 Label 41T8 Degree $41$ Order $1640$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{41}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(41, 8);

## Group action invariants

 Degree $n$: $41$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $8$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_{41}$ 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: (1,6,36,11,25,27,39,29,10,19,32,28,4,24,21,3,18,26,33,34,40,35,5,30,16,14,2,12,31,22,9,13,37,17,20,38,23,15,8,7), (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)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$8$:  $C_8$
$10$:  $C_{10}$
$20$:  20T1
$40$:  $C_{40}$

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 $2^{20},1$ $41$ $2$ $( 1,40)( 2,39)( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ 4A1 $4^{10},1$ $41$ $4$ $( 1, 9,40,32)( 2,18,39,23)( 3,27,38,14)( 4,36,37, 5)( 6,13,35,28)( 7,22,34,19)( 8,31,33,10)(11,17,30,24)(12,26,29,15)(16,21,25,20)$ 4A-1 $4^{10},1$ $41$ $4$ $( 1,32,40, 9)( 2,23,39,18)( 3,14,38,27)( 4, 5,37,36)( 6,28,35,13)( 7,19,34,22)( 8,10,33,31)(11,24,30,17)(12,15,29,26)(16,20,25,21)$ 5A1 $5^{8},1$ $41$ $5$ $( 1,10,18,16,37)( 2,20,36,32,33)( 3,30,13, 7,29)( 4,40,31,23,25)( 5, 9, 8,39,21)( 6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22)$ 5A-1 $5^{8},1$ $41$ $5$ $( 1,37,16,18,10)( 2,33,32,36,20)( 3,29, 7,13,30)( 4,25,23,31,40)( 5,21,39, 8, 9)( 6,17,14,26,19)(11,38,12,34,28)(15,22,35,24,27)$ 5A2 $5^{8},1$ $41$ $5$ $( 1,18,37,10,16)( 2,36,33,20,32)( 3,13,29,30, 7)( 4,31,25,40,23)( 5, 8,21, 9,39)( 6,26,17,19,14)(11,34,38,28,12)(15,24,22,27,35)$ 5A-2 $5^{8},1$ $41$ $5$ $( 1,16,10,37,18)( 2,32,20,33,36)( 3, 7,30,29,13)( 4,23,40,25,31)( 5,39, 9,21, 8)( 6,14,19,17,26)(11,12,28,38,34)(15,35,27,22,24)$ 8A1 $8^{5},1$ $41$ $8$ $( 1, 3, 9,27,40,38,32,14)( 2, 6,18,13,39,35,23,28)( 4,12,36,26,37,29, 5,15)( 7,21,22,25,34,20,19,16)( 8,24,31,11,33,17,10,30)$ 8A-1 $8^{5},1$ $41$ $8$ $( 1,14,32,38,40,27, 9, 3)( 2,28,23,35,39,13,18, 6)( 4,15, 5,29,37,26,36,12)( 7,16,19,20,34,25,22,21)( 8,30,10,17,33,11,31,24)$ 8A3 $8^{5},1$ $41$ $8$ $( 1,27,32, 3,40,14, 9,38)( 2,13,23, 6,39,28,18,35)( 4,26, 5,12,37,15,36,29)( 7,25,19,21,34,16,22,20)( 8,11,10,24,33,30,31,17)$ 8A-3 $8^{5},1$ $41$ $8$ $( 1,38, 9,14,40, 3,32,27)( 2,35,18,28,39, 6,23,13)( 4,29,36,15,37,12, 5,26)( 7,20,22,16,34,21,19,25)( 8,17,31,30,33,24,10,11)$ 10A1 $10^{4},1$ $41$ $10$ $( 1,31,18,25,37,40,10,23,16, 4)( 2,21,36, 9,33,39,20, 5,32, 8)( 3,11,13,34,29,38,30,28, 7,12)( 6,22,26,27,17,35,19,15,14,24)$ 10A-1 $10^{4},1$ $41$ $10$ $( 1, 4,16,23,10,40,37,25,18,31)( 2, 8,32, 5,20,39,33, 9,36,21)( 3,12, 7,28,30,38,29,34,13,11)( 6,24,14,15,19,35,17,27,26,22)$ 10A3 $10^{4},1$ $41$ $10$ $( 1,23,37,31,16,40,18, 4,10,25)( 2, 5,33,21,32,39,36, 8,20, 9)( 3,28,29,11, 7,38,13,12,30,34)( 6,15,17,22,14,35,26,24,19,27)$ 10A-3 $10^{4},1$ $41$ $10$ $( 1,25,10, 4,18,40,16,31,37,23)( 2, 9,20, 8,36,39,32,21,33, 5)( 3,34,30,12,13,38, 7,11,29,28)( 6,27,19,24,26,35,14,22,17,15)$ 20A1 $20^{2},1$ $41$ $20$ $( 1, 5,25, 2,10, 9, 4,20,18, 8,40,36,16,39,31,32,37,21,23,33)( 3,15,34, 6,30,27,12,19,13,24,38,26, 7,35,11,14,29,22,28,17)$ 20A-1 $20^{2},1$ $41$ $20$ $( 1,21,31,36,18, 9,25,33,37,39,40,20,10, 5,23,32,16, 8, 4, 2)( 3,22,11,26,13,27,34,17,29,35,38,19,30,15,28,14, 7,24,12, 6)$ 20A3 $20^{2},1$ $41$ $20$ $( 1, 8,23,20,37, 9,31, 2,16, 5,40,33,18,21, 4,32,10,39,25,36)( 3,24,28,19,29,27,11, 6, 7,15,38,17,13,22,12,14,30,35,34,26)$ 20A-3 $20^{2},1$ $41$ $20$ $( 1,39, 4,33,16, 9,23,36,10,21,40, 2,37, 8,25,32,18, 5,31,20)( 3,35,12,17, 7,27,28,26,30,22,38, 6,29,24,34,14,13,15,11,19)$ 20A7 $20^{2},1$ $41$ $20$ $( 1,20,31, 5,18,32,25, 8,37, 2,40,21,10,36,23, 9,16,33, 4,39)( 3,19,11,15,13,14,34,24,29, 6,38,22,30,26,28,27, 7,17,12,35)$ 20A-7 $20^{2},1$ $41$ $20$ $( 1,33,23,21,37,32,31,39,16,36,40, 8,18,20, 4, 9,10, 2,25, 5)( 3,17,28,22,29,14,11,35, 7,26,38,24,13,19,12,27,30, 6,34,15)$ 20A9 $20^{2},1$ $41$ $20$ $( 1,36,25,39,10,32, 4,21,18,33,40, 5,16, 2,31, 9,37,20,23, 8)( 3,26,34,35,30,14,12,22,13,17,38,15, 7, 6,11,27,29,19,28,24)$ 20A-9 $20^{2},1$ $41$ $20$ $( 1, 2, 4, 8,16,32,23, 5,10,20,40,39,37,33,25, 9,18,36,31,21)( 3, 6,12,24, 7,14,28,15,30,19,38,35,29,17,34,27,13,26,11,22)$ 40A1 $40,1$ $41$ $40$ $( 1,28, 5,17,25, 3, 2,15,10,34, 9, 6, 4,30,20,27,18,12, 8,19,40,13,36,24,16,38,39,26,31, 7,32,35,37,11,21,14,23,29,33,22)$ 40A-1 $40,1$ $41$ $40$ $( 1,19,33,12,23,27,21,30,37, 6,32,34,31,15,39, 3,16,17,36,28,40,22, 8,29,18,14,20,11, 4,35, 9, 7,10,26, 2,38,25,24, 5,13)$ 40A3 $40,1$ $41$ $40$ $( 1,15,20,13,31,14, 5,34,18,24,32,29,25, 6, 8,38,37,22, 2,30,40,26,21,28,10,27,36, 7,23,17, 9,12,16,35,33, 3, 4,19,39,11)$ 40A-3 $40,1$ $41$ $40$ $( 1,35,36,30,25,14,39,12,10,22,32,13, 4,17,21,38,18,15,33, 7,40, 6, 5,11,16,27, 2,29,31,19, 9,28,37,24,20, 3,23,26, 8,34)$ 40A7 $40,1$ $41$ $40$ $( 1,12,21, 6,31, 3,36,22,18,11, 9,26,25,13,33,27,37,34,39,17,40,29,20,35,10,38, 5,19,23,30,32,15,16,28, 8,14, 4, 7, 2,24)$ 40A-7 $40,1$ $41$ $40$ $( 1, 7, 8,15,23,38,20,17,37,13, 9,22,31,12, 2,14,16,30, 5,35,40,34,33,26,18, 3,21,24, 4,28,32,19,10,29,39,27,25,11,36, 6)$ 40A9 $40,1$ $41$ $40$ $( 1,22,33,29,23,14,21,11,37,35,32, 7,31,26,39,38,16,24,36,13,40,19, 8,12,18,27,20,30, 4, 6, 9,34,10,15, 2, 3,25,17, 5,28)$ 40A-9 $40,1$ $41$ $40$ $( 1,26,20,28,31,27, 5, 7,18,17,32,12,25,35, 8, 3,37,19, 2,11,40,15,21,13,10,14,36,34,23,24, 9,29,16, 6,33,38, 4,22,39,30)$ 40A11 $40,1$ $41$ $40$ $( 1,13, 5,24,25,38, 2,26,10, 7, 9,35, 4,11,20,14,18,29, 8,22,40,28,36,17,16, 3,39,15,31,34,32, 6,37,30,21,27,23,12,33,19)$ 40A-11 $40,1$ $41$ $40$ $( 1,34, 8,26,23, 3,20,24,37,28, 9,19,31,29, 2,27,16,11, 5, 6,40, 7,33,15,18,38,21,17, 4,13,32,22,10,12,39,14,25,30,36,35)$ 40A13 $40,1$ $41$ $40$ $( 1,30,39,22, 4,38,33, 6,16,29, 9,24,23,34,36,14,10,13,21,15,40,11, 2,19,37, 3, 8,35,25,12,32,17,18, 7, 5,27,31,28,20,26)$ 40A-13 $40,1$ $41$ $40$ $( 1,29,21,35,31,38,36,19,18,30, 9,15,25,28,33,14,37, 7,39,24,40,12,20, 6,10, 3, 5,22,23,11,32,26,16,13, 8,27, 4,34, 2,17)$ 40A17 $40,1$ $41$ $40$ $( 1, 6,36,11,25,27,39,29,10,19,32,28, 4,24,21, 3,18,26,33,34,40,35, 5,30,16,14, 2,12,31,22, 9,13,37,17,20,38,23,15, 8, 7)$ 40A-17 $40,1$ $41$ $40$ $( 1,24, 2, 7, 4,14, 8,28,16,15,32,30,23,19, 5,38,10,35,20,29,40,17,39,34,37,27,33,13,25,26, 9,11,18,22,36, 3,31, 6,21,12)$ 40A19 $40,1$ $41$ $40$ $( 1,11,39,19, 4, 3,33,35,16,12, 9,17,23, 7,36,27,10,28,21,26,40,30, 2,22,37,38, 8, 6,25,29,32,24,18,34, 5,14,31,13,20,15)$ 40A-19 $40,1$ $41$ $40$ $( 1,17, 2,34, 4,27, 8,13,16,26,32,11,23,22, 5, 3,10, 6,20,12,40,24,39, 7,37,14,33,28,25,15, 9,30,18,19,36,38,31,35,21,29)$ 41A $41$ $40$ $41$ $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $1640=2^{3} \cdot 5 \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 1640.47 magma: IdentifyGroup(G); Character table: 41 x 41 character table

magma: CharacterTable(G);