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Magma
magma: G := TransitiveGroup(21, 18);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^2:S_3$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,17,3,15,5,20,7,18,2,16,4,21,6,19)(8,13,11,9,14,12,10), (1,8,3,10,5,12,7,14,2,9,4,11,6,13)(15,17,19,21,16,18,20) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
14T15, 21T17, 42T56, 42T57, 42T62Siblings 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 |
$1^{21}$ | $1$ | $1$ | $()$ | |
$7^{2},1^{7}$ | $6$ | $7$ | $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ | |
$7^{2},1^{7}$ | $6$ | $7$ | $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ | |
$7^{2},1^{7}$ | $6$ | $7$ | $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ | |
$2^{7},1^{7}$ | $21$ | $2$ | $( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ | |
$7^{3}$ | $6$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ | |
$14,7$ | $21$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,15,14,16,13,17,12,18,11,19,10,20, 9,21)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ | |
$14,7$ | $21$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,15,13,17,11,19, 9,21,14,16,12,18,10,20)$ | |
$7^{3}$ | $6$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ | |
$14,7$ | $21$ | $14$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,15,12,18, 9,21,13,17,10,20,14,16,11,19)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$ | |
$7^{3}$ | $3$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ | |
$14,7$ | $21$ | $14$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,15,11,19,14,16,10,20,13,17, 9,21,12,18)$ | |
$14,7$ | $21$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,15,10,20,12,18,14,16, 9,21,11,19,13,17)$ | |
$14,7$ | $21$ | $14$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,15, 9,21,10,20,11,19,12,18,13,17,14,16)$ | |
$3^{7}$ | $98$ | $3$ | $( 1, 8,15)( 2, 9,21)( 3,10,20)( 4,11,19)( 5,12,18)( 6,13,17)( 7,14,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $294=2 \cdot 3 \cdot 7^{2}$ | 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: | 294.7 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 7A1 | 7A-1 | 7A2 | 7A-2 | 7A3 | 7A-3 | 7B1 | 7B-1 | 7C1 | 7C2 | 7C3 | 14A1 | 14A-1 | 14A3 | 14A-3 | 14A5 | 14A-5 | ||
Size | 1 | 21 | 98 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 21 | 21 | 21 | 21 | 21 | 21 | |
2 P | 1A | 1A | 3A | 7A-1 | 7A1 | 7A-2 | 7A3 | 7A-3 | 7A2 | 7B1 | 7B-1 | 7C2 | 7C3 | 7C1 | 7A3 | 7A1 | 7A-3 | 7A-2 | 7A-1 | 7A2 | |
3 P | 1A | 2A | 1A | 7A2 | 7A-2 | 7A-3 | 7A1 | 7A-1 | 7A3 | 7B-1 | 7B1 | 7C3 | 7C1 | 7C2 | 14A-5 | 14A3 | 14A5 | 14A1 | 14A-3 | 14A-1 | |
7 P | 1A | 2A | 3A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||||||
294.7.1a | R | ||||||||||||||||||||
294.7.1b | R | ||||||||||||||||||||
294.7.2a | R | ||||||||||||||||||||
294.7.3a1 | C | ||||||||||||||||||||
294.7.3a2 | C | ||||||||||||||||||||
294.7.3a3 | C | ||||||||||||||||||||
294.7.3a4 | C | ||||||||||||||||||||
294.7.3a5 | C | ||||||||||||||||||||
294.7.3a6 | C | ||||||||||||||||||||
294.7.3b1 | C | ||||||||||||||||||||
294.7.3b2 | C | ||||||||||||||||||||
294.7.3b3 | C | ||||||||||||||||||||
294.7.3b4 | C | ||||||||||||||||||||
294.7.3b5 | C | ||||||||||||||||||||
294.7.3b6 | C | ||||||||||||||||||||
294.7.6a1 | C | ||||||||||||||||||||
294.7.6a2 | C | ||||||||||||||||||||
294.7.6b1 | R | ||||||||||||||||||||
294.7.6b2 | R | ||||||||||||||||||||
294.7.6b3 | R |
magma: CharacterTable(G);