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Magma
magma: G := TransitiveGroup(15, 33);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^4:C_{10}$ | ||
CHM label: | $[3^{4}:2]5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,6,11)(4,14,9), (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15) | 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
Degree 3: None
Degree 5: $C_5$
Low degree siblings
15T33 x 7, 30T190 x 8, 45T118 x 2, 45T119 x 16, 45T120 x 8Siblings are shown 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 4,14, 9)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 2,12, 7)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2,12, 7)( 4,14, 9)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 4,14, 9)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 2,12, 7)( 4, 9,14)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1, 6,11)( 2, 7,12)( 4,14, 9)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2, 7,12)( 4, 9,14)( 5,15,10)$ |
$ 3, 3, 3, 3, 3 $ | $10$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$ |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $81$ | $2$ | $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$ |
$ 5, 5, 5 $ | $81$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 10, 5 $ | $81$ | $10$ | $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$ |
$ 5, 5, 5 $ | $81$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 10, 5 $ | $81$ | $10$ | $( 1,12, 3,14, 5, 6, 7, 8, 9,10)( 2,13, 4,15,11)$ |
$ 5, 5, 5 $ | $81$ | $5$ | $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
$ 10, 5 $ | $81$ | $10$ | $( 1, 8, 5, 2, 9,11,13,15, 7, 4)( 3,10,12,14, 6)$ |
$ 5, 5, 5 $ | $81$ | $5$ | $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
$ 10, 5 $ | $81$ | $10$ | $( 1,15,14,13,12,11, 5, 9, 3, 7)( 2, 6,10, 4, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $810=2 \cdot 3^{4} \cdot 5$ | 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: | 810.102 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . . . . . . . 1 1 1 1 1 1 1 1 1 3 4 4 4 4 4 4 4 4 4 . . . . . . . . . 5 1 . . . . . . . . 1 1 1 1 1 1 1 1 1 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 5a 10a 5b 10b 5c 10c 5d 10d 2P 1a 3a 3b 3c 3d 3e 3f 3g 3h 1a 5b 5b 5c 5c 5d 5d 5a 5a 3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a 5d 10d 5a 10a 5b 10b 5c 10c 5P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 1a 2a 1a 2a 1a 2a 1a 2a 7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a 5b 10b 5c 10c 5d 10d 5a 10a X.1 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 -1 X.3 1 1 1 1 1 1 1 1 1 -1 A -A B -B /A -/A /B -/B X.4 1 1 1 1 1 1 1 1 1 -1 B -B /A -/A /B -/B A -A X.5 1 1 1 1 1 1 1 1 1 -1 /B -/B A -A B -B /A -/A X.6 1 1 1 1 1 1 1 1 1 -1 /A -/A /B -/B A -A B -B X.7 1 1 1 1 1 1 1 1 1 1 A A B B /A /A /B /B X.8 1 1 1 1 1 1 1 1 1 1 B B /A /A /B /B A A X.9 1 1 1 1 1 1 1 1 1 1 /B /B A A B B /A /A X.10 1 1 1 1 1 1 1 1 1 1 /A /A /B /B A A B B X.11 10 -5 1 4 -2 -2 1 4 -2 . . . . . . . . . X.12 10 1 -5 -2 4 1 -2 4 -2 . . . . . . . . . X.13 10 1 -2 4 -2 1 -5 -2 4 . . . . . . . . . X.14 10 4 -2 -2 -5 -2 4 1 1 . . . . . . . . . X.15 10 4 4 1 1 -2 -2 -2 -5 . . . . . . . . . X.16 10 -2 1 -2 4 -5 1 -2 4 . . . . . . . . . X.17 10 -2 4 -5 -2 4 -2 1 1 . . . . . . . . . X.18 10 -2 -2 1 1 4 4 -5 -2 . . . . . . . . . A = E(5)^4 B = E(5)^3 |
magma: CharacterTable(G);