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Magma
magma: G := TransitiveGroup(21, 4);
Group invariants
Abstract group: | $F_7$ | magma: IdentifyGroup(G);
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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 | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,9,5,18,10,14)(2,8,6,16,12,13)(3,7,4,17,11,15)(19,21,20)$, $(1,4,8,12,15,18,19)(2,5,7,11,14,16,20)(3,6,9,10,13,17,21)$ | 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
Degree 3: $C_3$
Degree 7: $F_7$
Low degree siblings
7T4, 14T4, 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 | $1^{21}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{9},1^{3}$ | $7$ | $2$ | $9$ | $( 4,19)( 5,20)( 6,21)( 7,16)( 8,18)( 9,17)(10,13)(11,14)(12,15)$ |
3A1 | $3^{7}$ | $7$ | $3$ | $14$ | $( 1, 3, 2)( 4,13, 7)( 5,15, 9)( 6,14, 8)(10,16,19)(11,18,21)(12,17,20)$ |
3A-1 | $3^{7}$ | $7$ | $3$ | $14$ | $( 1, 2, 3)( 4, 7,13)( 5, 9,15)( 6, 8,14)(10,19,16)(11,21,18)(12,20,17)$ |
6A1 | $6^{3},3$ | $7$ | $6$ | $17$ | $( 1, 2, 3)( 4,16,13,19, 7,10)( 5,17,15,20, 9,12)( 6,18,14,21, 8,11)$ |
6A-1 | $6^{3},3$ | $7$ | $6$ | $17$ | $( 1, 3, 2)( 4,10, 7,19,13,16)( 5,12, 9,20,15,17)( 6,11, 8,21,14,18)$ |
7A | $7^{3}$ | $6$ | $7$ | $18$ | $( 1,15, 4,18, 8,19,12)( 2,14, 5,16, 7,20,11)( 3,13, 6,17, 9,21,10)$ |
Malle's constant $a(G)$: $1/9$
magma: ConjugacyClasses(G);
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);
Regular extensions
Data not computed