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Magma
magma: G := TransitiveGroup(21, 41);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^4.S_3^2$ | ||
| 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: | (1,8,4,11,2,9)(3,10,5,12,6,13)(7,14)(15,18,19,17,21,20), (1,10,18,2,9,16,3,8,21,4,14,19,5,13,17,6,12,15,7,11,20) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $42$: $F_7$ $126$: 21T10 $882$: 14T26 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
21T41 x 5, 42T465 x 6, 42T472 x 3, 42T475 x 2Siblings 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 | $2^{10},1$ | $147$ | $2$ | $10$ | $( 1, 5)( 2, 4)( 6, 7)( 8,18)( 9,17)(10,16)(11,15)(12,21)(13,20)(14,19)$ |
| 3A | $3^{7}$ | $98$ | $3$ | $14$ | $( 1,14,18)( 2,10,21)( 3,13,17)( 4, 9,20)( 5,12,16)( 6, 8,19)( 7,11,15)$ |
| 3B1 | $3^{7}$ | $98$ | $3$ | $14$ | $( 1, 9,20)( 2,14,19)( 3,12,18)( 4,10,17)( 5, 8,16)( 6,13,15)( 7,11,21)$ |
| 3B-1 | $3^{7}$ | $98$ | $3$ | $14$ | $( 1, 9,15)( 2, 8,20)( 3,14,18)( 4,13,16)( 5,12,21)( 6,11,19)( 7,10,17)$ |
| 3C1 | $3^{6},1^{3}$ | $343$ | $3$ | $12$ | $( 1, 2, 6)( 4, 7, 5)( 8,14,10)( 9,11,12)(15,21,17)(16,18,19)$ |
| 3C-1 | $3^{6},1^{3}$ | $343$ | $3$ | $12$ | $( 1, 6, 2)( 4, 5, 7)( 8,10,14)( 9,12,11)(15,17,21)(16,19,18)$ |
| 6A1 | $6^{3},2,1$ | $1029$ | $6$ | $16$ | $( 1, 7, 2, 5, 6, 4)( 8,16,14,18,10,19)( 9,21,11,17,12,15)(13,20)$ |
| 6A-1 | $6^{3},2,1$ | $1029$ | $6$ | $16$ | $( 1, 4, 6, 5, 2, 7)( 8,19,10,18,14,16)( 9,15,12,17,11,21)(13,20)$ |
| 7A | $7^{3}$ | $6$ | $7$ | $18$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| 7B1 | $7^{3}$ | $6$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$ |
| 7B-1 | $7^{3}$ | $6$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
| 7C1 | $7^{2},1^{7}$ | $9$ | $7$ | $12$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)$ |
| 7C-1 | $7^{2},1^{7}$ | $9$ | $7$ | $12$ | $( 1, 4, 7, 3, 6, 2, 5)(15,20,18,16,21,19,17)$ |
| 7D | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
| 7E | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 6, 4, 2, 7, 5, 3)(15,19,16,20,17,21,18)$ |
| 7F | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ |
| 7G | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ |
| 7H | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 3, 5, 7, 2, 4, 6)(15,21,20,19,18,17,16)$ |
| 7I | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$ |
| 7J | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ |
| 7K1 | $7,1^{14}$ | $18$ | $7$ | $6$ | $( 8,13,11, 9,14,12,10)$ |
| 7K-1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$ |
| 7L1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$ |
| 7L-1 | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$ |
| 7M1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ |
| 7M-1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ |
| 7N1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
| 7N-1 | $7^{3}$ | $18$ | $7$ | $18$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| 7O1 | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)$ |
| 7O-1 | $7^{2},1^{7}$ | $18$ | $7$ | $12$ | $( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
| 14A1 | $14,2^{3},1$ | $441$ | $14$ | $16$ | $( 2, 7)( 3, 6)( 4, 5)( 8,16,10,21,12,19,14,17, 9,15,11,20,13,18)$ |
| 14A-1 | $14,2^{3},1$ | $441$ | $14$ | $16$ | $( 1, 6)( 2, 5)( 3, 4)( 8,15,11,19,14,16,10,20,13,17, 9,21,12,18)$ |
| 21A1 | $21$ | $294$ | $21$ | $20$ | $( 1,13,16, 7,10,20, 6,14,17, 5,11,21, 4, 8,18, 3,12,15, 2, 9,19)$ |
| 21A2 | $21$ | $294$ | $21$ | $20$ | $( 1,13,19, 6, 8,16, 4,10,20, 2,12,17, 7,14,21, 5, 9,18, 3,11,15)$ |
| 21B1 | $21$ | $294$ | $21$ | $20$ | $( 1,13,18, 3, 9,16, 5,12,21, 7, 8,19, 2,11,17, 4,14,15, 6,10,20)$ |
| 21B-1 | $21$ | $294$ | $21$ | $20$ | $( 1,11,19, 5, 9,17, 2,14,15, 6,12,20, 3,10,18, 7, 8,16, 4,13,21)$ |
| 21B2 | $21$ | $294$ | $21$ | $20$ | $( 1, 8,17, 4, 9,21, 7,10,18, 3,11,15, 6,12,19, 2,13,16, 5,14,20)$ |
| 21B-2 | $21$ | $294$ | $21$ | $20$ | $( 1,14,20, 2,13,18, 3,12,16, 4,11,21, 5,10,19, 6, 9,17, 7, 8,15)$ |
Malle's constant $a(G)$: $1/6$
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.bk | magma: IdentifyGroup(G);
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| Character table: | 39 x 39 character table |
magma: CharacterTable(G);