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Magma
magma: G := TransitiveGroup(41, 6);
Group action invariants
Degree $n$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{41}:C_{10}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (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), (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) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $C_{10}$ 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
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31, 39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$ |
$ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8, 39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$ |
$ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14, 39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$ |
$ 10, 10, 10, 10, 1 $ | $41$ | $10$ | $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30, 39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $41$ | $2$ | $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$ |
$ 41 $ | $10$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ |
$ 41 $ | $10$ | $41$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ |
$ 41 $ | $10$ | $41$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$ |
$ 41 $ | $10$ | $41$ | $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $410=2 \cdot 5 \cdot 41$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 410.1 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 1 1 1 1 . . . . 5 1 1 1 1 1 1 1 1 1 1 . . . . 41 1 . . . . . . . . . 1 1 1 1 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41a 41b 41c 41d 2P 1a 5b 5c 5a 5d 5d 5a 5c 5b 1a 41b 41a 41d 41c 3P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41c 41d 41b 41a 5P 1a 2a 1a 1a 1a 2a 2a 2a 1a 2a 41b 41a 41d 41c 7P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41c 41d 41b 41a 11P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41c 41d 41b 41a 13P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41c 41d 41b 41a 17P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41d 41c 41a 41b 19P 1a 10d 5d 5c 5b 10c 10b 10a 5a 2a 41d 41c 41a 41b 23P 1a 10b 5b 5d 5a 10d 10a 10c 5c 2a 41a 41b 41c 41d 29P 1a 10d 5d 5c 5b 10c 10b 10a 5a 2a 41c 41d 41b 41a 31P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 41a 41b 41c 41d 37P 1a 10c 5c 5a 5d 10a 10d 10b 5b 2a 41a 41b 41c 41d 41P 1a 10a 5a 5b 5c 10b 10c 10d 5d 2a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 X.3 1 A -/A -B -/B /B B /A -A -1 1 1 1 1 X.4 1 B -/B -/A -A A /A /B -B -1 1 1 1 1 X.5 1 /B -B -A -/A /A A B -/B -1 1 1 1 1 X.6 1 /A -A -/B -B B /B A -/A -1 1 1 1 1 X.7 1 -/A -A -/B -B -B -/B -A -/A 1 1 1 1 1 X.8 1 -/B -B -A -/A -/A -A -B -/B 1 1 1 1 1 X.9 1 -B -/B -/A -A -A -/A -/B -B 1 1 1 1 1 X.10 1 -A -/A -B -/B -/B -B -/A -A 1 1 1 1 1 X.11 10 . . . . . . . . . C D F E X.12 10 . . . . . . . . . D C E F X.13 10 . . . . . . . . . E F C D X.14 10 . . . . . . . . . F E D C A = -E(5) B = -E(5)^2 C = E(41)^3+E(41)^7+E(41)^11+E(41)^12+E(41)^13+E(41)^28+E(41)^29+E(41)^30+E(41)^34+E(41)^38 D = E(41)^6+E(41)^14+E(41)^15+E(41)^17+E(41)^19+E(41)^22+E(41)^24+E(41)^26+E(41)^27+E(41)^35 E = E(41)+E(41)^4+E(41)^10+E(41)^16+E(41)^18+E(41)^23+E(41)^25+E(41)^31+E(41)^37+E(41)^40 F = E(41)^2+E(41)^5+E(41)^8+E(41)^9+E(41)^20+E(41)^21+E(41)^32+E(41)^33+E(41)^36+E(41)^39 |
magma: CharacterTable(G);