Show commands:
Magma
magma: G := TransitiveGroup(7, 4);
Group action invariants
Degree $n$: | $7$ | 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: | $F_7$ | ||
CHM label: | $F_{42}(7) = 7:6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | yes | 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: | (1,3,2,6,4,5), (1,2,3,4,5,6,7) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
14T4, 21T4, 42T4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{7}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{3},1$ | $7$ | $2$ | $3$ | $(2,7)(3,6)(4,5)$ |
3A1 | $3^{2},1$ | $7$ | $3$ | $4$ | $(2,5,3)(4,6,7)$ |
3A-1 | $3^{2},1$ | $7$ | $3$ | $4$ | $(2,3,5)(4,7,6)$ |
6A1 | $6,1$ | $7$ | $6$ | $5$ | $(2,6,5,7,3,4)$ |
6A-1 | $6,1$ | $7$ | $6$ | $5$ | $(2,4,3,7,5,6)$ |
7A | $7$ | $6$ | $7$ | $6$ | $(1,5,2,6,3,7,4)$ |
Malle's constant $a(G)$: $1/3$
magma: ConjugacyClasses(G);
Group invariants
Order: | $42=2 \cdot 3 \cdot 7$ | 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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Nilpotency class: | not nilpotent | ||
Label: | 42.1 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 7A | ||
Size | 1 | 7 | 7 | 7 | 7 | 7 | 6 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 7A | |
3 P | 1A | 2A | 1A | 1A | 2A | 2A | 7A | |
7 P | 1A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 1A | |
Type | ||||||||
42.1.1a | R | |||||||
42.1.1b | R | |||||||
42.1.1c1 | C | |||||||
42.1.1c2 | C | |||||||
42.1.1d1 | C | |||||||
42.1.1d2 | C | |||||||
42.1.6a | R |
magma: CharacterTable(G);