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Magma
magma: G := TransitiveGroup(42, 19);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $19$ | 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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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15,41)(2,16,42)(3,18,37)(4,17,38)(5,14,39)(6,13,40)(7,27,24)(8,28,23)(9,29,20)(10,30,19)(11,26,21)(12,25,22)(31,33,35)(32,34,36), (1,9,13,21,25,33,37,4,8,16,19,27,32,39)(2,10,14,22,26,34,38,3,7,15,20,28,31,40)(5,12,17,23,29,36,42,6,11,18,24,30,35,41) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$
Degree 21: 21T11
Low degree siblings
21T11, 42T23Siblings 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, 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, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,14,26)( 8,13,25)( 9,16,27)(10,15,28)(11,17,29)(12,18,30)(19,37,32) (20,38,31)(21,39,33)(22,40,34)(23,41,36)(24,42,35)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,26,14)( 8,25,13)( 9,27,16)(10,28,15)(11,29,17)(12,30,18)(19,32,37) (20,31,38)(21,33,39)(22,34,40)(23,36,41)(24,35,42)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23) (22,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,41)(40,42)$ | |
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 5)( 4, 6)( 7,13,26, 8,14,25)( 9,18,27,12,16,30)(10,17,28,11,15,29) (19,38,32,20,37,31)(21,41,33,23,39,36)(22,42,34,24,40,35)$ | |
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 5)( 4, 6)( 7,25,14, 8,26,13)( 9,30,16,12,27,18)(10,29,15,11,28,17) (19,31,37,20,32,38)(21,36,39,23,33,41)(22,35,40,24,34,42)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,22,23) (20,21,24)(25,28,30)(26,27,29)(31,33,35)(32,34,36)(37,40,41)(38,39,42)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,16,29)( 8,15,30)( 9,17,26)(10,18,25)(11,14,27) (12,13,28)(19,40,36)(20,39,35)(21,42,31)(22,41,32)(23,37,34)(24,38,33)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,27,17)( 8,28,18)( 9,29,14)(10,30,13)(11,26,16) (12,25,15)(19,34,41)(20,33,42)(21,35,38)(22,36,37)(23,32,40)(24,31,39)$ | |
$ 14, 14, 14 $ | $9$ | $14$ | $( 1, 7,13,20,25,31,37, 2, 8,14,19,26,32,38)( 3,11,15,24,28,35,40, 5,10,17,22, 29,34,42)( 4,12,16,23,27,36,39, 6, 9,18,21,30,33,41)$ | |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 8,13,19,25,32,37)( 2, 7,14,20,26,31,38)( 3,10,15,22,28,34,40) ( 4, 9,16,21,27,33,39)( 5,11,17,24,29,35,42)( 6,12,18,23,30,36,41)$ | |
$ 21, 21 $ | $6$ | $21$ | $( 1,10,18,19,28,36,37, 3,12,13,22,30,32,40, 6, 8,15,23,25,34,41) ( 2, 9,17,20,27,35,38, 4,11,14,21,29,31,39, 5, 7,16,24,26,33,42)$ | |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,19,37,13,32, 8,25)( 2,20,38,14,31, 7,26)( 3,22,40,15,34,10,28) ( 4,21,39,16,33, 9,27)( 5,24,42,17,35,11,29)( 6,23,41,18,36,12,30)$ | |
$ 14, 14, 14 $ | $9$ | $14$ | $( 1,20,37,14,32, 7,25, 2,19,38,13,31, 8,26)( 3,24,40,17,34,11,28, 5,22,42,15, 35,10,29)( 4,23,39,18,33,12,27, 6,21,41,16,36, 9,30)$ | |
$ 21, 21 $ | $6$ | $21$ | $( 1,22,41,13,34,12,25, 3,23,37,15,36, 8,28, 6,19,40,18,32,10,30) ( 2,21,42,14,33,11,26, 4,24,38,16,35, 7,27, 5,20,39,17,31, 9,29)$ |
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: | 126.8 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 3B1 | 3B-1 | 3C1 | 3C-1 | 6A1 | 6A-1 | 7A1 | 7A-1 | 14A1 | 14A-1 | 21A1 | 21A-1 | ||
Size | 1 | 3 | 2 | 7 | 7 | 14 | 14 | 21 | 21 | 3 | 3 | 9 | 9 | 6 | 6 | |
2 P | 1A | 1A | 3A | 3B-1 | 3B1 | 3C-1 | 3C1 | 3B1 | 3B-1 | 7A1 | 7A-1 | 7A-1 | 7A1 | 21A1 | 21A-1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 7A-1 | 7A1 | 14A-1 | 14A1 | 7A1 | 7A-1 | |
7 P | 1A | 2A | 3A | 3B1 | 3B-1 | 3C1 | 3C-1 | 6A1 | 6A-1 | 1A | 1A | 2A | 2A | 3A | 3A | |
Type | ||||||||||||||||
126.8.1a | R | |||||||||||||||
126.8.1b | R | |||||||||||||||
126.8.1c1 | C | |||||||||||||||
126.8.1c2 | C | |||||||||||||||
126.8.1d1 | C | |||||||||||||||
126.8.1d2 | C | |||||||||||||||
126.8.2a | R | |||||||||||||||
126.8.2b1 | C | |||||||||||||||
126.8.2b2 | C | |||||||||||||||
126.8.3a1 | C | |||||||||||||||
126.8.3a2 | C | |||||||||||||||
126.8.3b1 | C | |||||||||||||||
126.8.3b2 | C | |||||||||||||||
126.8.6a1 | C | |||||||||||||||
126.8.6a2 | C |
magma: CharacterTable(G);