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Magma
magma: G := TransitiveGroup(40, 43);
Group action invariants
Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{20}.C_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,21,3,23,2,22,4,24)(5,36,39,10,6,35,40,9)(7,34,38,11,8,33,37,12)(13,18,32,25,14,17,31,26)(15,20,29,27,16,19,30,28), (1,39,9,16,2,40,10,15)(3,37,11,13,4,38,12,14)(5,25,7,28,6,26,8,27)(17,31,35,22,18,32,36,21)(19,30,34,24,20,29,33,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $16$: $C_8:C_2$ $20$: $F_5$ $40$: $F_{5}\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $F_5$
Degree 8: $C_8:C_2$
Degree 10: $F_5$
Degree 20: 20T9
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, 1, 1, 1, 1 $ | $10$ | $2$ | $( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,36)(12,35)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,26)(20,25)(21,22)(23,24)$ |
$ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,39,38,40)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 3, 2, 4)( 5,39, 6,40)( 7,38, 8,37)( 9,36,10,35)(11,33,12,34)(13,32,14,31) (15,29,16,30)(17,26,18,25)(19,28,20,27)(21,23,22,24)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 4, 2, 3)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,34,12,33)(13,31,14,32) (15,30,16,29)(17,25,18,26)(19,27,20,28)(21,24,22,23)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 5,18,13, 2, 6,17,14)( 3, 8,20,15, 4, 7,19,16)( 9,29,11,31,10,30,12,32) (21,25,37,34,22,26,38,33)(23,27,40,36,24,28,39,35)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 5,34,29, 2, 6,33,30)( 3, 8,36,31, 4, 7,35,32)( 9,22,25,15,10,21,26,16) (11,24,27,14,12,23,28,13)(17,40,20,37,18,39,19,38)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 7,18,16, 2, 8,17,15)( 3, 5,20,13, 4, 6,19,14)( 9,32,11,29,10,31,12,30) (21,27,37,36,22,28,38,35)(23,26,40,33,24,25,39,34)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1, 7,34,32, 2, 8,33,31)( 3, 5,36,29, 4, 6,35,30)( 9,23,25,13,10,24,26,14) (11,22,27,15,12,21,28,16)(17,37,20,39,18,38,19,40)$ |
$ 20, 20 $ | $4$ | $20$ | $( 1, 9,20,28,36, 4,12,18,26,34, 2,10,19,27,35, 3,11,17,25,33)( 5,13,23,29,38, 8,15,22,31,39, 6,14,24,30,37, 7,16,21,32,40)$ |
$ 20, 20 $ | $4$ | $20$ | $( 1,10,20,27,36, 3,12,17,26,33, 2, 9,19,28,35, 4,11,18,25,34)( 5,14,23,30,38, 7,15,21,31,40, 6,13,24,29,37, 8,16,22,32,39)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,11,19,26,36)( 2,12,20,25,35)( 3,10,18,28,33)( 4, 9,17,27,34) ( 5,16,24,31,38)( 6,15,23,32,37)( 7,14,22,29,40)( 8,13,21,30,39)$ |
$ 10, 10, 10, 10 $ | $4$ | $10$ | $( 1,12,19,25,36, 2,11,20,26,35)( 3, 9,18,27,33, 4,10,17,28,34)( 5,15,24,32,38, 6,16,23,31,37)( 7,13,22,30,40, 8,14,21,29,39)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $80=2^{4} \cdot 5$ | 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: | 80.29 | magma: IdentifyGroup(G);
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Character table: |
2 4 3 4 3 4 4 3 3 3 3 2 2 2 2 5 1 . 1 1 . . . . . . 1 1 1 1 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 20a 20b 5a 10a 2P 1a 1a 1a 2b 2b 2b 4b 4c 4b 4c 10a 10a 5a 5a 3P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20a 20b 5a 10a 5P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 4a 4a 1a 2b 7P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20a 20b 5a 10a 11P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20b 20a 5a 10a 13P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 20b 20a 5a 10a 17P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 20b 20a 5a 10a 19P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20b 20a 5a 10a 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 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 X.5 1 -1 1 1 -1 -1 B -B B -B 1 1 1 1 X.6 1 -1 1 1 -1 -1 -B B -B B 1 1 1 1 X.7 1 1 1 -1 -1 -1 B B -B -B -1 -1 1 1 X.8 1 1 1 -1 -1 -1 -B -B B B -1 -1 1 1 X.9 2 . -2 . A -A . . . . . . 2 -2 X.10 2 . -2 . -A A . . . . . . 2 -2 X.11 4 . 4 -4 . . . . . . 1 1 -1 -1 X.12 4 . 4 4 . . . . . . -1 -1 -1 -1 X.13 4 . -4 . . . . . . . C -C -1 1 X.14 4 . -4 . . . . . . . -C C -1 1 A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i C = -E(20)-E(20)^9+E(20)^13+E(20)^17 = -Sqrt(-5) = -i5 |
magma: CharacterTable(G);