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Magma
magma: G := TransitiveGroup(21, 42);
Group invariants
| Abstract group: | $C_7^3:C_{18}$ | magma: IdentifyGroup(G);
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| Order: | $6174=2 \cdot 3^{2} \cdot 7^{3}$ | 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 $t$: | $42$ | 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,2,5,7,6,3)(8,9,12,14,13,10)(15,20,21,17,19,18)$, $(1,14,18,6,9,17,7,8,21,3,12,19,5,10,20,4,11,16)(2,13,15)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $9$: $C_9$ $18$: $C_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T42 x 18, 42T466 x 19Siblings 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 | $2^{9},1^{3}$ | $343$ | $2$ | $9$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ |
| 3A1 | $3^{6},1^{3}$ | $343$ | $3$ | $12$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| 3A-1 | $3^{6},1^{3}$ | $343$ | $3$ | $12$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
| 6A1 | $6^{3},1^{3}$ | $343$ | $6$ | $15$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ |
| 6A-1 | $6^{3},1^{3}$ | $343$ | $6$ | $15$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ |
| 7A | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$ |
| 7B | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
| 7C | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ |
| 7D | $7,1^{14}$ | $18$ | $7$ | $6$ | $( 8,14,13,12,11,10, 9)$ |
| 7E | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ |
| 7F | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 4, 7, 3, 6, 2, 5)(15,20,18,16,21,19,17)$ |
| 7G | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 5, 2, 6, 3, 7, 4)(15,19,16,20,17,21,18)$ |
| 7H | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$ |
| 7I | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$ |
| 7J | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| 7K | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)$ |
| 7L | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ |
| 7M | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 2, 3, 4, 5, 6, 7)(15,21,20,19,18,17,16)$ |
| 7N | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ |
| 7O | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$ |
| 7P | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
| 7Q | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| 7R | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$ |
| 7S | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)$ |
| 9A1 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1,15, 8)( 2,17, 9, 3,19,10, 5,16,12)( 4,21,11, 7,20,14, 6,18,13)$ |
| 9A-1 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1, 8,15)( 2,10,17, 5, 9,16, 3,12,19)( 4,14,21, 6,11,18, 7,13,20)$ |
| 9A2 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1,15, 8)( 2,16,10, 3,17,12, 5,19, 9)( 4,18,14, 7,21,13, 6,20,11)$ |
| 9A-2 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1, 8,15)( 2, 9,19, 5,12,17, 3,10,16)( 4,11,20, 6,13,21, 7,14,18)$ |
| 9A4 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1, 8,15)( 2,12,16, 5,10,19, 3, 9,17)( 4,13,18, 6,14,20, 7,11,21)$ |
| 9A-4 | $9^{2},3$ | $343$ | $9$ | $18$ | $( 1,15, 8)( 2,19,12, 3,16, 9, 5,17,10)( 4,20,13, 7,18,11, 6,21,14)$ |
| 18A1 | $18,3$ | $343$ | $18$ | $19$ | $( 1,15, 8)( 2,20, 9, 6,19,13, 5,21,12, 7,17,14, 3,18,10, 4,16,11)$ |
| 18A-1 | $18,3$ | $343$ | $18$ | $19$ | $( 1, 8,15)( 2,13,17, 4, 9,21, 3,11,19, 7,10,20, 5,14,16, 6,12,18)$ |
| 18A5 | $18,3$ | $343$ | $18$ | $19$ | $( 1, 8,15)( 2,14,19, 4,12,20, 3,13,16, 7, 9,18, 5,11,17, 6,10,21)$ |
| 18A-5 | $18,3$ | $343$ | $18$ | $19$ | $( 1, 8,15)( 2,11,16, 4,10,18, 3,14,17, 7,12,21, 5,13,19, 6, 9,20)$ |
| 18A7 | $18,3$ | $343$ | $18$ | $19$ | $( 1,15, 8)( 2,18,12, 6,16,14, 5,20,10, 7,19,11, 3,21, 9, 4,17,13)$ |
| 18A-7 | $18,3$ | $343$ | $18$ | $19$ | $( 1,15, 8)( 2,21,10, 6,17,11, 5,18, 9, 7,16,13, 3,20,12, 4,19,14)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
37 x 37 character tablemagma: CharacterTable(G);