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Magma
magma: G := TransitiveGroup(42, 48);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $48$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_3\times C_{21}):C_4$ | ||
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: | (1,15,26,39,9,21,31,3,14,27,37,8,20,33)(2,13,25,38,7,19,32)(4,16,30,40,12,22,34)(5,17,28,42,11,23,35,6,18,29,41,10,24,36), (1,12,2,10)(3,11)(4,7,6,9)(5,8)(13,42,14,40)(15,41)(16,38,17,37)(18,39)(19,36,20,34)(21,35)(22,32,23,31)(24,33)(25,29,26,30)(27,28) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $14$: $D_{7}$ $28$: 28T3 $36$: $C_3^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $C_3^2:C_4$
Degree 7: $D_{7}$
Degree 14: $D_{7}$
Degree 21: None
Low degree siblings
42T48Siblings 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, 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 $ | $4$ | $3$ | $( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,29,30)(34,35,36)(40,41,42)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 5, 6)( 7, 8)(10,11)(13,15)(17,18)(19,21)(23,24)(25,27)(28,29)(32,33) (35,36)(38,39)(41,42)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ | $63$ | $4$ | $( 1, 4)( 2, 5, 3, 6)( 7,41, 8,42)( 9,40)(10,38,11,39)(12,37)(13,35,15,36) (14,34)(16,31)(17,32,18,33)(19,28,21,29)(20,30)(22,26)(23,25,24,27)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ | $63$ | $4$ | $( 1, 4)( 2, 6, 3, 5)( 7,42, 8,41)( 9,40)(10,39,11,38)(12,37)(13,36,15,35) (14,34)(16,31)(17,33,18,32)(19,29,21,28)(20,30)(22,26)(23,27,24,25)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,10,18,22,29,35,40, 6,11,16,23,28,34,42, 5,12,17,24,30,36,41)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,11,17,22,28,36,40, 5,10,16,24,29,34,41, 6,12,18,23,30,35,42)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,12,16,22,30,34,40)( 5,11,18,24,28,35,41)( 6,10,17,23,29,36,42)$ |
$ 14, 14, 7, 7 $ | $18$ | $14$ | $( 1, 7,14,19,26,32,37, 2, 9,13,20,25,31,38)( 3, 8,15,21,27,33,39) ( 4,10,16,23,30,36,40, 6,12,17,22,29,34,42)( 5,11,18,24,28,35,41)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1, 9,14,20,26,31,37)( 2, 7,13,19,25,32,38)( 3, 8,15,21,27,33,39) ( 4,10,18,22,29,35,40, 6,11,16,23,28,34,42, 5,12,17,24,30,36,41)$ |
$ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 9,14,20,26,31,37)( 2, 7,13,19,25,32,38)( 3, 8,15,21,27,33,39) ( 4,12,16,22,30,34,40)( 5,11,18,24,28,35,41)( 6,10,17,23,29,36,42)$ |
$ 14, 14, 7, 7 $ | $18$ | $14$ | $( 1,13,26,38, 9,19,31, 2,14,25,37, 7,20,32)( 3,15,27,39, 8,21,33) ( 4,16,30,40,12,22,34)( 5,17,28,42,11,23,35, 6,18,29,41,10,24,36)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,16,30,40,12,22,34)( 5,18,28,41,11,24,35)( 6,17,29,42,10,23,36)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,17,28,40,10,24,34, 6,18,30,42,11,22,36, 5,16,29,41,12,23,35)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,18,29,40,11,23,34, 5,17,30,41,10,22,35, 6,16,28,42,12,24,36)$ |
$ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,14,26,37, 9,20,31)( 2,13,25,38, 7,19,32)( 3,15,27,39, 8,21,33) ( 4,16,30,40,12,22,34)( 5,18,28,41,11,24,35)( 6,17,29,42,10,23,36)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1,14,26,37, 9,20,31)( 2,13,25,38, 7,19,32)( 3,15,27,39, 8,21,33) ( 4,17,28,40,10,24,34, 6,18,30,42,11,22,36, 5,16,29,41,12,23,35)$ |
$ 14, 14, 7, 7 $ | $18$ | $14$ | $( 1,19,37,13,31, 7,26, 2,20,38,14,32, 9,25)( 3,21,39,15,33, 8,27) ( 4,22,40,16,34,12,30)( 5,23,41,17,35,10,28, 6,24,42,18,36,11,29)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,22,40,16,34,12,30)( 5,24,41,18,35,11,28)( 6,23,42,17,36,10,29)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,23,41,16,36,11,30, 6,24,40,17,35,12,29, 5,22,42,18,34,10,28)$ |
$ 21, 21 $ | $4$ | $21$ | $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,24,42,16,35,10,30, 5,23,40,18,36,12,28, 6,22,41,17,34,11,29)$ |
$ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,20,37,14,31, 9,26)( 2,19,38,13,32, 7,25)( 3,21,39,15,33, 8,27) ( 4,22,40,16,34,12,30)( 5,24,41,18,35,11,28)( 6,23,42,17,36,10,29)$ |
$ 21, 7, 7, 7 $ | $4$ | $21$ | $( 1,20,37,14,31, 9,26)( 2,19,38,13,32, 7,25)( 3,21,39,15,33, 8,27) ( 4,23,41,16,36,11,30, 6,24,40,17,35,12,29, 5,22,42,18,34,10,28)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | 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: | 252.32 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);