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Magma
magma: G := TransitiveGroup(42, 21);
Group action invariants
| Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times D_{21}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $21$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,36,2,34,3,35)(4,33,5,31,6,32)(7,29,8,30,9,28)(10,26,12,25,11,27)(13,24,15,23,14,22)(16,21,18,20,17,19)(37,41,38,42,39,40), (1,16,2,18,3,17)(4,13,5,15,6,14)(7,12,8,11,9,10)(19,41,21,42,20,40)(22,38,24,39,23,37)(25,35,27,36,26,34)(28,32,29,33,30,31) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $14$: $D_{7}$ $18$: $S_3\times C_3$ $42$: 21T3, $D_{21}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 7: $D_{7}$
Degree 14: $D_{7}$
Degree 21: None
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,29,30)(34,35,36)(40,41,42)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $( 4, 6, 5)(10,11,12)(16,17,18)(22,23,24)(28,30,29)(34,36,35)(40,42,41)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 4)( 2, 5)( 3, 6)( 7,42)( 8,40)( 9,41)(10,39)(11,38)(12,37)(13,35)(14,34) (15,36)(16,31)(17,33)(18,32)(19,29)(20,28)(21,30)(22,27)(23,25)(24,26)$ | |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4, 2, 5, 3, 6)( 7,42, 8,40, 9,41)(10,37,12,38,11,39)(13,35,15,36,14,34) (16,32,18,33,17,31)(19,29,21,30,20,28)(22,26,24,25,23,27)$ | |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4, 3, 6, 2, 5)( 7,42, 9,41, 8,40)(10,38,11,37,12,39)(13,35,14,34,15,36) (16,33,17,32,18,31)(19,29,20,28,21,30)(22,25,23,26,24,27)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$ | |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$ | |
| $ 21, 21 $ | $2$ | $21$ | $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$ | |
| $ 21, 7, 7, 7 $ | $2$ | $21$ | $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$ |
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: | 126.13 | magma: IdentifyGroup(G);
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| Character table: | 36 x 36 character table |
magma: CharacterTable(G);