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Magma
magma: G := TransitiveGroup(40, 4);
Group action invariants
Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{10}:C_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $40$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11,18,28,36,3,10,19,25,34)(2,12,17,27,35,4,9,20,26,33)(5,16,24,32,38,7,13,21,29,39)(6,15,23,31,37,8,14,22,30,40), (1,38,4,40)(2,37,3,39)(5,33,8,36)(6,34,7,35)(9,30,11,32)(10,29,12,31)(13,27,15,25)(14,28,16,26)(17,23,19,21)(18,24,20,22) | 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, $C_2^2$ $8$: $C_4\times C_2$ $10$: $D_{5}$ $20$: $D_{10}$, 20T2 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 5: $D_{5}$
Degree 8: $C_4\times C_2$
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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,35)(34,36)(37,40)(38,39)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)(37,39)(38,40)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,37,11,39)(10,38,12,40)(13,33,15,36)(14,34,16,35) (17,30,19,32)(18,29,20,31)(21,26,23,28)(22,25,24,27)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 6, 4, 7)( 2, 5, 3, 8)( 9,38,11,40)(10,37,12,39)(13,34,15,35)(14,33,16,36) (17,29,19,31)(18,30,20,32)(21,25,23,27)(22,26,24,28)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,40,11,38)(10,39,12,37)(13,35,15,34)(14,36,16,33) (17,31,19,29)(18,32,20,30)(21,27,23,25)(22,28,24,26)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $5$ | $4$ | $( 1, 8, 4, 5)( 2, 7, 3, 6)( 9,39,11,37)(10,40,12,38)(13,36,15,33)(14,35,16,34) (17,32,19,30)(18,31,20,29)(21,28,23,26)(22,27,24,25)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1, 9,18,26,36, 2,10,17,25,35)( 3,12,19,27,34, 4,11,20,28,33)( 5,14,24,30,38, 6,13,23,29,37)( 7,15,21,31,39, 8,16,22,32,40)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,18,25,36)( 2, 9,17,26,35)( 3,11,19,28,34)( 4,12,20,27,33) ( 5,13,24,29,38)( 6,14,23,30,37)( 7,16,21,32,39)( 8,15,22,31,40)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,11,18,28,36, 3,10,19,25,34)( 2,12,17,27,35, 4, 9,20,26,33)( 5,16,24,32,38, 7,13,21,29,39)( 6,15,23,31,37, 8,14,22,30,40)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,12,18,27,36, 4,10,20,25,33)( 2,11,17,28,35, 3, 9,19,26,34)( 5,15,24,31,38, 8,13,22,29,40)( 6,16,23,32,37, 7,14,21,30,39)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,17,36, 9,25, 2,18,35,10,26)( 3,20,34,12,28, 4,19,33,11,27)( 5,23,38,14,29, 6,24,37,13,30)( 7,22,39,15,32, 8,21,40,16,31)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,18,36,10,25)( 2,17,35, 9,26)( 3,19,34,11,28)( 4,20,33,12,27) ( 5,24,38,13,29)( 6,23,37,14,30)( 7,21,39,16,32)( 8,22,40,15,31)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,19,36,11,25, 3,18,34,10,28)( 2,20,35,12,26, 4,17,33, 9,27)( 5,21,38,16,29, 7,24,39,13,32)( 6,22,37,15,30, 8,23,40,14,31)$ |
$ 10, 10, 10, 10 $ | $2$ | $10$ | $( 1,20,36,12,25, 4,18,33,10,27)( 2,19,35,11,26, 3,17,34, 9,28)( 5,22,38,15,29, 8,24,40,13,31)( 6,21,37,16,30, 7,23,39,14,32)$ |
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: | 40.7 | 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 2b 2c 4a 4b 4c 4d 10a 5a 10b 10c 10d 5b 10e 10f 2P 1a 1a 1a 1a 2c 2c 2c 2c 5b 5b 5b 5b 5a 5a 5a 5a 3P 1a 2a 2b 2c 4d 4c 4b 4a 10d 5b 10e 10f 10a 5a 10b 10c 5P 1a 2a 2b 2c 4a 4b 4c 4d 2a 1a 2b 2c 2a 1a 2b 2c 7P 1a 2a 2b 2c 4d 4c 4b 4a 10d 5b 10e 10f 10a 5a 10b 10c 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 -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 X.5 1 -1 1 -1 A -A A -A -1 1 1 -1 -1 1 1 -1 X.6 1 -1 1 -1 -A A -A A -1 1 1 -1 -1 1 1 -1 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 -2 2 . . . . B -B B -B *B -*B *B -*B X.10 2 -2 -2 2 . . . . *B -*B *B -*B B -B B -B X.11 2 -2 2 -2 . . . . B -B -B B *B -*B -*B *B X.12 2 -2 2 -2 . . . . *B -*B -*B *B B -B -B B X.13 2 2 -2 -2 . . . . -*B -*B *B *B -B -B B B X.14 2 2 -2 -2 . . . . -B -B B B -*B -*B *B *B X.15 2 2 2 2 . . . . -*B -*B -*B -*B -B -B -B -B X.16 2 2 2 2 . . . . -B -B -B -B -*B -*B -*B -*B A = -E(4) = -Sqrt(-1) = -i B = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);