Show commands:
Magma
magma: G := TransitiveGroup(7, 3);
Group action invariants
Degree $n$: | $7$ | 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: | $C_7:C_3$ | ||
CHM label: | $F_{21}(7) = 7:3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
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Nilpotency class: | $-1$ (not nilpotent) | magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,4)(3,6,5), (1,2,3,4,5,6,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
21T2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1 $ | $7$ | $3$ | $(2,3,5)(4,7,6)$ |
$ 3, 3, 1 $ | $7$ | $3$ | $(2,5,3)(4,6,7)$ |
$ 7 $ | $3$ | $7$ | $(1,2,3,4,5,6,7)$ |
$ 7 $ | $3$ | $7$ | $(1,4,7,3,6,2,5)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $21=3 \cdot 7$ | 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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Label: | 21.1 | magma: IdentifyGroup(G);
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Character table: |
3 1 1 1 . . 7 1 . . 1 1 1a 3a 3b 7a 7b 2P 1a 3b 3a 7a 7b 3P 1a 1a 1a 7b 7a 5P 1a 3b 3a 7b 7a 7P 1a 3a 3b 1a 1a X.1 1 1 1 1 1 X.2 1 A /A 1 1 X.3 1 /A A 1 1 X.4 3 . . B /B X.5 3 . . /B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)+E(7)^2+E(7)^4 = (-1+Sqrt(-7))/2 = b7 |
magma: CharacterTable(G);