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Magma
magma: G := TransitiveGroup(21, 10);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{21}:C_6$ | ||
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: | (1,3,2)(4,15,8)(5,13,9)(6,14,7)(10,17,20)(11,18,21)(12,16,19), (1,10,17,15,5,19)(2,12,18,14,6,21)(3,11,16,13,4,20)(8,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: $F_7$
Low degree siblings
42T18, 42T22Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $7$ | $3$ | $( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,18)(11,20,16)(12,21,17)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $7$ | $3$ | $( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,18,19)(11,16,20)(12,17,21)$ |
$ 6, 6, 6, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4,10, 7,19,13,18)( 5,12, 8,21,14,17)( 6,11, 9,20,15,16)$ |
$ 6, 6, 6, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4,18,13,19, 7,10)( 5,17,14,21, 8,12)( 6,16,15,20, 9,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $21$ | $2$ | $( 2, 3)( 4,19)( 5,21)( 6,20)( 7,18)( 8,17)( 9,16)(10,13)(11,15)(12,14)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 2, 3)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,20,17)(11,21,18)(12,19,16)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 2, 3)( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,16,21)(11,17,19)(12,18,20)$ |
$ 21 $ | $6$ | $21$ | $( 1, 4, 9,12,13,18,21, 3, 6, 8,11,15,17,20, 2, 5, 7,10,14,16,19)$ |
$ 7, 7, 7 $ | $6$ | $7$ | $( 1, 5, 8,12,14,17,21)( 2, 6, 9,10,15,18,19)( 3, 4, 7,11,13,16,20)$ |
$ 21 $ | $6$ | $21$ | $( 1, 6, 7,12,15,16,21, 2, 4, 8,10,13,17,19, 3, 5, 9,11,14,18,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $126=2 \cdot 3^{2} \cdot 7$ | 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 | ||
Label: | 126.9 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 . . . . . . 3 2 2 2 1 1 1 2 2 2 1 1 1 7 1 . . . . . 1 . . 1 1 1 1a 3a 3b 6a 6b 2a 3c 3d 3e 21a 7a 21b 2P 1a 3b 3a 3a 3b 1a 3c 3e 3d 21b 7a 21a 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 7a 7a 7a 5P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21a 7a 21b 7P 1a 3a 3b 6a 6b 2a 3c 3d 3e 3c 1a 3c 11P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21b 7a 21a 13P 1a 3a 3b 6a 6b 2a 3c 3d 3e 21b 7a 21a 17P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21a 7a 21b 19P 1a 3a 3b 6a 6b 2a 3c 3d 3e 21b 7a 21a X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 1 1 1 X.3 1 A /A -/A -A -1 1 A /A 1 1 1 X.4 1 /A A -A -/A -1 1 /A A 1 1 1 X.5 1 A /A /A A 1 1 A /A 1 1 1 X.6 1 /A A A /A 1 1 /A A 1 1 1 X.7 2 2 2 . . . -1 -1 -1 -1 2 -1 X.8 2 B /B . . . -1 -A -/A -1 2 -1 X.9 2 /B B . . . -1 -/A -A -1 2 -1 X.10 6 . . . . . 6 . . -1 -1 -1 X.11 6 . . . . . -3 . . C -1 *C X.12 6 . . . . . -3 . . *C -1 C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19 = (1-Sqrt(21))/2 = -b21 |
magma: CharacterTable(G);