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Magma
magma: G := TransitiveGroup(40, 49);
Group action invariants
Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:\SD_{16}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $10$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,33,3,35,2,34,4,36)(5,31,8,30,6,32,7,29)(9,27,11,25,10,28,12,26)(13,24,15,22,14,23,16,21)(17,40,19,38,18,39,20,37), (1,15,7,17,11,3,13,5,20,10)(2,16,8,18,12,4,14,6,19,9)(21,36,27,39,32)(22,35,28,40,31)(23,33,26,37,30,24,34,25,38,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $10$: $D_{5}$ $16$: $QD_{16}$ $20$: $D_{10}$ $40$: 20T7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $D_{5}$
Degree 8: $QD_{16}$
Degree 10: $D_5$
Degree 20: 20T11
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(15,16)(17,18)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32) (33,35)(34,36)(37,40)(38,39)$ |
$ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,28,26,27)(29,31,30,32)(33,35,34,36)(37,40,38,39)$ |
$ 10, 10, 10, 5, 5 $ | $4$ | $10$ | $( 1, 5,11,15,20, 3, 7,10,13,17)( 2, 6,12,16,19, 4, 8, 9,14,18)(21,27,32,36,39) (22,28,31,35,40)(23,25,30,33,38,24,26,29,34,37)$ |
$ 20, 20 $ | $4$ | $20$ | $( 1, 5,12,16,20, 3, 8, 9,13,17, 2, 6,11,15,19, 4, 7,10,14,18)(21,25,31,34,39, 24,28,30,36,37,22,26,32,33,40,23,27,29,35,38)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,11,13,20)( 2, 8,12,14,19)( 3, 5,10,15,17)( 4, 6, 9,16,18) (21,27,32,36,39)(22,28,31,35,40)(23,26,30,34,38)(24,25,29,33,37)$ |
$ 10, 10, 10, 5, 5 $ | $4$ | $10$ | $( 1, 7,11,13,20)( 2, 8,12,14,19)( 3, 6,10,16,17, 4, 5, 9,15,18) (21,26,32,34,39,23,27,30,36,38)(22,25,31,33,40,24,28,29,35,37)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1, 8,11,14,20, 2, 7,12,13,19)( 3, 6,10,16,17, 4, 5, 9,15,18)(21,28,32,35,39, 22,27,31,36,40)(23,25,30,33,38,24,26,29,34,37)$ |
$ 20, 20 $ | $4$ | $20$ | $( 1, 9,19, 5,13, 4,12,17, 7,16, 2,10,20, 6,14, 3,11,18, 8,15)(21,30,40,25,36, 23,31,37,27,34,22,29,39,26,35,24,32,38,28,33)$ |
$ 10, 10, 10, 5, 5 $ | $4$ | $10$ | $( 1, 9,20, 6,13, 4,11,18, 7,16)( 2,10,19, 5,14, 3,12,17, 8,15)(21,31,39,28,36, 22,32,40,27,35)(23,30,38,26,34)(24,29,37,25,33)$ |
$ 10, 10, 10, 5, 5 $ | $4$ | $10$ | $( 1,11,20, 7,13)( 2,12,19, 8,14)( 3, 9,17, 6,15, 4,10,18, 5,16) (21,30,39,26,36,23,32,38,27,34)(22,29,40,25,35,24,31,37,28,33)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,20, 7,13)( 2,12,19, 8,14)( 3,10,17, 5,15)( 4, 9,18, 6,16) (21,32,39,27,36)(22,31,40,28,35)(23,30,38,26,34)(24,29,37,25,33)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,12,20, 8,13, 2,11,19, 7,14)( 3, 9,17, 6,15, 4,10,18, 5,16)(21,31,39,28,36, 22,32,40,27,35)(23,29,38,25,34,24,30,37,26,33)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $20$ | $4$ | $( 1,21, 2,22)( 3,23, 4,24)( 5,38, 6,37)( 7,39, 8,40)( 9,33,10,34)(11,36,12,35) (13,32,14,31)(15,30,16,29)(17,26,18,25)(19,28,20,27)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1,21, 4,23, 2,22, 3,24)( 5,37, 7,39, 6,38, 8,40)( 9,34,12,35,10,33,11,36) (13,32,16,30,14,31,15,29)(17,25,20,27,18,26,19,28)$ |
$ 8, 8, 8, 8, 8 $ | $10$ | $8$ | $( 1,22, 4,24, 2,21, 3,23)( 5,38, 7,40, 6,37, 8,39)( 9,33,12,36,10,34,11,35) (13,31,16,29,14,32,15,30)(17,26,20,28,18,25,19,27)$ |
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: | 80.16 | magma: IdentifyGroup(G);
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Character table: |
2 4 2 4 3 2 2 3 2 3 2 2 2 3 3 2 3 3 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . . 1a 2a 2b 4a 10a 20a 5a 10b 10c 20b 10d 10e 5b 10f 4b 8a 8b 2P 1a 1a 1a 2b 5b 10f 5b 5b 5b 10c 5a 5a 5a 5a 2b 4a 4a 3P 1a 2a 2b 4a 10e 20b 5b 10d 10f 20a 10a 10b 5a 10c 4b 8a 8b 5P 1a 2a 2b 4a 2a 4a 1a 2a 2b 4a 2a 2a 1a 2b 4b 8b 8a 7P 1a 2a 2b 4a 10d 20b 5b 10e 10f 20a 10b 10a 5a 10c 4b 8b 8a 11P 1a 2a 2b 4a 10a 20a 5a 10b 10c 20b 10d 10e 5b 10f 4b 8a 8b 13P 1a 2a 2b 4a 10e 20b 5b 10d 10f 20a 10a 10b 5a 10c 4b 8b 8a 17P 1a 2a 2b 4a 10d 20b 5b 10e 10f 20a 10b 10a 5a 10c 4b 8a 8b 19P 1a 2a 2b 4a 10b 20a 5a 10a 10c 20b 10e 10d 5b 10f 4b 8a 8b X.1 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 1 X.3 1 -1 1 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 -1 -1 -1 X.5 2 . 2 -2 . -2 2 . 2 -2 . . 2 2 . . . X.6 2 -2 2 2 A -A -A A -A -*A *A *A -*A -*A . . . X.7 2 -2 2 2 *A -*A -*A *A -*A -A A A -A -A . . . X.8 2 . -2 . . . 2 . -2 . . . 2 -2 . E -E X.9 2 . -2 . . . 2 . -2 . . . 2 -2 . -E E X.10 2 . 2 -2 B A -A -B -A *A C -C -*A -*A . . . X.11 2 . 2 -2 C *A -*A -C -*A A -B B -A -A . . . X.12 2 . 2 -2 -C *A -*A C -*A A B -B -A -A . . . X.13 2 . 2 -2 -B A -A B -A *A -C C -*A -*A . . . X.14 2 2 2 2 -*A -*A -*A -*A -*A -A -A -A -A -A . . . X.15 2 2 2 2 -A -A -A -A -A -*A -*A -*A -*A -*A . . . X.16 4 . -4 . . . D . -D . . . *D -*D . . . X.17 4 . -4 . . . *D . -*D . . . D -D . . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -E(5)+E(5)^4 C = -E(5)^2+E(5)^3 D = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 E = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 |
magma: CharacterTable(G);