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Magma
magma: G := TransitiveGroup(40, 3);
Group action invariants
Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:C_8$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $40$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,27,10,33,17,3,26,11,35,19,2,28,9,34,18,4,25,12,36,20)(5,31,14,40,23,8,30,15,37,22,6,32,13,39,24,7,29,16,38,21), (1,5,3,8,2,6,4,7)(9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35)(17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ $10$: $D_{5}$ $20$: 20T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $D_{5}$
Degree 8: $C_8$
Degree 10: $D_5$
Degree 20: 20T2
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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,39,38,40)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,36,34,35)(37,40,38,39)$ |
$ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 5, 3, 8, 2, 6, 4, 7)( 9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35) (17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28)$ |
$ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 6, 3, 7, 2, 5, 4, 8)( 9,38,12,40,10,37,11,39)(13,33,16,35,14,34,15,36) (17,30,19,32,18,29,20,31)(21,26,23,28,22,25,24,27)$ |
$ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 7, 4, 6, 2, 8, 3, 5)( 9,40,11,38,10,39,12,37)(13,35,15,33,14,36,16,34) (17,32,20,30,18,31,19,29)(21,28,24,26,22,27,23,25)$ |
$ 8, 8, 8, 8, 8 $ | $5$ | $8$ | $( 1, 8, 4, 5, 2, 7, 3, 6)( 9,39,11,37,10,40,12,38)(13,36,15,34,14,35,16,33) (17,31,20,29,18,32,19,30)(21,27,24,25,22,28,23,26)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 9,17,25,35)( 2,10,18,26,36)( 3,12,19,27,34)( 4,11,20,28,33) ( 5,13,23,29,37)( 6,14,24,30,38)( 7,15,21,32,40)( 8,16,22,31,39)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,10,17,26,35, 2, 9,18,25,36)( 3,11,19,28,34, 4,12,20,27,33)( 5,14,23,30,37, 6,13,24,29,38)( 7,16,21,31,40, 8,15,22,32,39)$ |
$ 20, 20 $ | $2$ | $20$ | $( 1,11,18,27,35, 4,10,19,25,33, 2,12,17,28,36, 3, 9,20,26,34)( 5,15,24,31,37, 7,14,22,29,40, 6,16,23,32,38, 8,13,21,30,39)$ |
$ 20, 20 $ | $2$ | $20$ | $( 1,12,18,28,35, 3,10,20,25,34, 2,11,17,27,36, 4, 9,19,26,33)( 5,16,24,32,37, 8,14,21,29,39, 6,15,23,31,38, 7,13,22,30,40)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,17,35, 9,25)( 2,18,36,10,26)( 3,19,34,12,27)( 4,20,33,11,28) ( 5,23,37,13,29)( 6,24,38,14,30)( 7,21,40,15,32)( 8,22,39,16,31)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,18,35,10,25, 2,17,36, 9,26)( 3,20,34,11,27, 4,19,33,12,28)( 5,24,37,14,29, 6,23,38,13,30)( 7,22,40,16,32, 8,21,39,15,31)$ |
$ 20, 20 $ | $2$ | $20$ | $( 1,19,36,11,25, 3,18,33, 9,27, 2,20,35,12,26, 4,17,34,10,28)( 5,22,38,15,29, 8,24,40,13,31, 6,21,37,16,30, 7,23,39,14,32)$ |
$ 20, 20 $ | $2$ | $20$ | $( 1,20,36,12,25, 4,18,34, 9,28, 2,19,35,11,26, 3,17,33,10,27)( 5,21,38,16,29, 7,24,39,13,32, 6,22,37,15,30, 8,23,40,14,31)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $40=2^{3} \cdot 5$ | 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: | 40.1 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 5 1 1 1 1 . . . . 1 1 1 1 1 1 1 1 1a 2a 4a 4b 8a 8b 8c 8d 5a 10a 20a 20b 5b 10b 20c 20d 2P 1a 1a 2a 2a 4a 4a 4b 4b 5b 5b 10b 10b 5a 5a 10a 10a 3P 1a 2a 4b 4a 8d 8c 8b 8a 5b 10b 20c 20d 5a 10a 20a 20b 5P 1a 2a 4a 4b 8b 8a 8d 8c 1a 2a 4b 4a 1a 2a 4a 4b 7P 1a 2a 4b 4a 8c 8d 8a 8b 5b 10b 20c 20d 5a 10a 20a 20b 11P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c 13P 1a 2a 4a 4b 8b 8a 8d 8c 5b 10b 20d 20c 5a 10a 20b 20a 17P 1a 2a 4a 4b 8a 8b 8c 8d 5b 10b 20d 20c 5a 10a 20b 20a 19P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c X.1 1 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 1 1 X.3 1 -1 A -A C -C /C -/C 1 -1 -A A 1 -1 A -A X.4 1 -1 A -A -C C -/C /C 1 -1 -A A 1 -1 A -A X.5 1 -1 -A A -/C /C -C C 1 -1 A -A 1 -1 -A A X.6 1 -1 -A A /C -/C C -C 1 -1 A -A 1 -1 -A A X.7 1 1 -1 -1 A A -A -A 1 1 -1 -1 1 1 -1 -1 X.8 1 1 -1 -1 -A -A A A 1 1 -1 -1 1 1 -1 -1 X.9 2 -2 B -B . . . . D -D E -E *D -*D -F F X.10 2 -2 B -B . . . . *D -*D F -F D -D -E E X.11 2 -2 -B B . . . . D -D -E E *D -*D F -F X.12 2 -2 -B B . . . . *D -*D -F F D -D E -E X.13 2 2 -2 -2 . . . . D D -D -D *D *D -*D -*D X.14 2 2 -2 -2 . . . . *D *D -*D -*D D D -D -D X.15 2 2 2 2 . . . . D D D D *D *D *D *D X.16 2 2 2 2 . . . . *D *D *D *D D D D D A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i C = -E(8)^3 D = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 E = E(20)^13+E(20)^17 F = E(20)+E(20)^9 |
magma: CharacterTable(G);