# Properties

 Label 41T5 Degree $41$ Order $328$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{41}:C_{8}$

Show commands: Magma

magma: G := TransitiveGroup(41, 5);

## Group action invariants

 Degree $n$: $41$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{41}:C_{8}$ 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: (1,27,32,3,40,14,9,38)(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), (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)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

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 $2^{20},1$ $41$ $2$ $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$ 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 $4^{10},1$ $41$ $4$ $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20)( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$ 8A1 $8^{5},1$ $41$ $8$ $( 2,28,33, 4,41,15,10,39)( 3,14,24, 7,40,29,19,36)( 5,27, 6,13,38,16,37,30)( 8,26,20,22,35,17,23,21)( 9,12,11,25,34,31,32,18)$ 8A-1 $8^{5},1$ $41$ $8$ $( 2,39,10,15,41, 4,33,28)( 3,36,19,29,40, 7,24,14)( 5,30,37,16,38,13, 6,27)( 8,21,23,17,35,22,20,26)( 9,18,32,31,34,25,11,12)$ 8A3 $8^{5},1$ $41$ $8$ $( 2, 4,10,28,41,39,33,15)( 3, 7,19,14,40,36,24,29)( 5,13,37,27,38,30, 6,16)( 8,22,23,26,35,21,20,17)( 9,25,32,12,34,18,11,31)$ 8A-3 $8^{5},1$ $41$ $8$ $( 2,15,33,39,41,28,10, 4)( 3,29,24,36,40,14,19, 7)( 5,16, 6,30,38,27,37,13)( 8,17,20,21,35,26,23,22)( 9,31,11,18,34,12,32,25)$ 41A1 $41$ $8$ $41$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ 41A2 $41$ $8$ $41$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ 41A4 $41$ $8$ $41$ $( 1,25, 8,32,15,39,22, 5,29,12,36,19, 2,26, 9,33,16,40,23, 6,30,13,37,20, 3,27,10,34,17,41,24, 7,31,14,38,21, 4,28,11,35,18)$ 41A7 $41$ $8$ $41$ $( 1,22, 2,23, 3,24, 4,25, 5,26, 6,27, 7,28, 8,29, 9,30,10,31,11,32,12,33,13,34,14,35,15,36,16,37,17,38,18,39,19,40,20,41,21)$ 41A8 $41$ $8$ $41$ $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15,19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $328=2^{3} \cdot 41$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 328.12 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 41A1 41A2 41A4 41A7 41A8 Size 1 41 41 41 41 41 41 41 8 8 8 8 8 2 P 1A 1A 2A 2A 4A1 4A-1 4A-1 4A1 41A8 41A4 41A1 41A2 41A7 41 P 1A 2A 4A1 4A-1 8A-3 8A3 8A-1 8A1 41A7 41A8 41A2 41A4 41A1 Type 328.12.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 328.12.1b R $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ 328.12.1c1 C $1$ $1$ $−1$ $−1$ $−i$ $i$ $i$ $−i$ $1$ $1$ $1$ $1$ $1$ 328.12.1c2 C $1$ $1$ $−1$ $−1$ $i$ $−i$ $−i$ $i$ $1$ $1$ $1$ $1$ $1$ 328.12.1d1 C $1$ $−1$ $−ζ82$ $ζ82$ $ζ83$ $−ζ8$ $ζ8$ $−ζ83$ $1$ $1$ $1$ $1$ $1$ 328.12.1d2 C $1$ $−1$ $ζ82$ $−ζ82$ $−ζ8$ $ζ83$ $−ζ83$ $ζ8$ $1$ $1$ $1$ $1$ $1$ 328.12.1d3 C $1$ $−1$ $−ζ82$ $ζ82$ $−ζ83$ $ζ8$ $−ζ8$ $ζ83$ $1$ $1$ $1$ $1$ $1$ 328.12.1d4 C $1$ $−1$ $ζ82$ $−ζ82$ $ζ8$ $−ζ83$ $ζ83$ $−ζ8$ $1$ $1$ $1$ $1$ $1$ 328.12.8a1 R $8$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−20+ζ41−19+ζ41−16+ζ41−7+ζ417+ζ4116+ζ4119+ζ4120$ $ζ41−14+ζ41−9+ζ41−3+ζ41−1+ζ41+ζ413+ζ419+ζ4114$ $ζ41−18+ζ41−13+ζ41−6+ζ41−2+ζ412+ζ416+ζ4113+ζ4118$ $ζ41−17+ζ41−11+ζ41−10+ζ41−8+ζ418+ζ4110+ζ4111+ζ4117$ $ζ41−15+ζ41−12+ζ41−5+ζ41−4+ζ414+ζ415+ζ4112+ζ4115$ 328.12.8a2 R $8$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−17+ζ41−11+ζ41−10+ζ41−8+ζ418+ζ4110+ζ4111+ζ4117$ $ζ41−20+ζ41−19+ζ41−16+ζ41−7+ζ417+ζ4116+ζ4119+ζ4120$ $ζ41−14+ζ41−9+ζ41−3+ζ41−1+ζ41+ζ413+ζ419+ζ4114$ $ζ41−15+ζ41−12+ζ41−5+ζ41−4+ζ414+ζ415+ζ4112+ζ4115$ $ζ41−18+ζ41−13+ζ41−6+ζ41−2+ζ412+ζ416+ζ4113+ζ4118$ 328.12.8a3 R $8$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−18+ζ41−13+ζ41−6+ζ41−2+ζ412+ζ416+ζ4113+ζ4118$ $ζ41−15+ζ41−12+ζ41−5+ζ41−4+ζ414+ζ415+ζ4112+ζ4115$ $ζ41−17+ζ41−11+ζ41−10+ζ41−8+ζ418+ζ4110+ζ4111+ζ4117$ $ζ41−14+ζ41−9+ζ41−3+ζ41−1+ζ41+ζ413+ζ419+ζ4114$ $ζ41−20+ζ41−19+ζ41−16+ζ41−7+ζ417+ζ4116+ζ4119+ζ4120$ 328.12.8a4 R $8$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−15+ζ41−12+ζ41−5+ζ41−4+ζ414+ζ415+ζ4112+ζ4115$ $ζ41−17+ζ41−11+ζ41−10+ζ41−8+ζ418+ζ4110+ζ4111+ζ4117$ $ζ41−20+ζ41−19+ζ41−16+ζ41−7+ζ417+ζ4116+ζ4119+ζ4120$ $ζ41−18+ζ41−13+ζ41−6+ζ41−2+ζ412+ζ416+ζ4113+ζ4118$ $ζ41−14+ζ41−9+ζ41−3+ζ41−1+ζ41+ζ413+ζ419+ζ4114$ 328.12.8a5 R $8$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ41−14+ζ41−9+ζ41−3+ζ41−1+ζ41+ζ413+ζ419+ζ4114$ $ζ41−18+ζ41−13+ζ41−6+ζ41−2+ζ412+ζ416+ζ4113+ζ4118$ $ζ41−15+ζ41−12+ζ41−5+ζ41−4+ζ414+ζ415+ζ4112+ζ4115$ $ζ41−20+ζ41−19+ζ41−16+ζ41−7+ζ417+ζ4116+ζ4119+ζ4120$ $ζ41−17+ζ41−11+ζ41−10+ζ41−8+ζ418+ζ4110+ζ4111+ζ4117$

magma: CharacterTable(G);