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Magma
magma: G := TransitiveGroup(21, 42);
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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| Group: | $C_7^3:C_{18}$ | ||
| 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: | (1,2,5,7,6,3)(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);
|
Low degree resolvents
|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 | Representative |
| $1^{21}$ | $1$ | $1$ | $()$ | |
| $7,1^{14}$ | $18$ | $7$ | $(1,4,7,3,6,2,5)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)$ | |
| $7^{2},1^{7}$ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$ | |
| $7^{3}$ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$ | |
| $3^{6},1^{3}$ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ | |
| $3^{6},1^{3}$ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ | |
| $6^{3},1^{3}$ | $343$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ | |
| $2^{9},1^{3}$ | $343$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ | |
| $6^{3},1^{3}$ | $343$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1,18, 9, 7,21,12, 5,20,11)( 2,15,13)( 3,19,10, 4,16,14, 6,17, 8)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1,21, 9, 6,19,12, 2,15,11)( 3,16,13)( 4,17, 8, 5,18,10, 7,20,14)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1,20,13, 5,21,10, 6,16,11)( 2,15,14, 7,18,12, 3,17, 8)( 4,19, 9)$ | |
| $18,3$ | $343$ | $18$ | $( 1,17, 8, 7,19,14, 2,15, 9, 5,16,12, 6,21,13, 4,18,11)( 3,20,10)$ | |
| $18,3$ | $343$ | $18$ | $( 1,19,13, 7,16, 9, 2,15,10, 5,17, 8, 6,20,12, 4,21,11)( 3,18,14)$ | |
| $18,3$ | $343$ | $18$ | $( 1,16,10, 2,15,12, 7,17, 8, 4,20, 9, 3,21,14, 5,19,11)( 6,18,13)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1, 9,21, 5,11,18, 7,12,20)( 2,13,15)( 3,10,16, 6, 8,19, 4,14,17)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1,10,17, 5,14,19, 7, 9,20)( 2,11,21)( 3,12,18, 6, 8,16, 4,13,15)$ | |
| $9^{2},3$ | $343$ | $9$ | $( 1,12,21, 3, 9,18, 4,11,20)( 2,14,16, 7,10,19, 6, 8,17)( 5,13,15)$ | |
| $18,3$ | $343$ | $18$ | $( 1,11,17, 7,13,15, 4,12,16, 2, 9,19, 3,14,21, 6, 8,20)( 5,10,18)$ | |
| $18,3$ | $343$ | $18$ | $( 1,14,20)( 2,10,21, 4, 9,16, 3,13,15, 7,11,19, 5,12,17, 6, 8,18)$ | |
| $18,3$ | $343$ | $18$ | $( 1,13,15, 6, 8,21, 7,14,18, 3,11,16, 5, 9,17, 4,10,20)( 2,12,19)$ |
magma: ConjugacyClasses(G);
Group invariants
| 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 | ||
| Label: | 6174.n | magma: IdentifyGroup(G);
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| Character table: | 37 x 37 character table |
magma: CharacterTable(G);