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Magma
magma: G := TransitiveGroup(30, 26);
Group action invariants
| Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_6\times F_5$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| Nilpotency class: | $-1$ (not nilpotent) | magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2)(3,27,9,15)(4,28,10,16)(5,23,17,30)(6,24,18,29)(7,20,25,13)(8,19,26,14)(11,12)(21,22), (1,10,5,7,21,29,25,27,11,19,15,17)(2,9,6,8,22,30,26,28,12,20,16,18)(3,23,13)(4,24,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $20$: $F_5$ $24$: 24T2 $40$: $F_{5}\times C_2$ $60$: $F_5\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 5: $F_5$
Degree 6: $C_6$
Degree 10: $F_{5}\times C_2$
Degree 15: $F_5\times C_3$
Low degree siblings
30T26Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3, 9)( 4,10)( 5,17)( 6,18)( 7,25)( 8,26)(13,20)(14,19)(15,27)(16,28)(23,30) (24,29)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 3,16, 9,28)( 4,15,10,27)( 5,29,17,24)( 6,30,18,23)( 7,14,25,19)( 8,13,26,20)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 3,28, 9,16)( 4,27,10,15)( 5,24,17,29)( 6,23,18,30)( 7,19,25,14)( 8,20,26,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(11,12)(13,19)(14,20)(15,28) (16,27)(21,22)(23,29)(24,30)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $5$ | $4$ | $( 1, 2)( 3,15, 9,27)( 4,16,10,28)( 5,30,17,23)( 6,29,18,24)( 7,13,25,20) ( 8,14,26,19)(11,12)(21,22)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $5$ | $4$ | $( 1, 2)( 3,27, 9,15)( 4,28,10,16)( 5,23,17,30)( 6,24,18,29)( 7,20,25,13) ( 8,19,26,14)(11,12)(21,22)$ |
| $ 30 $ | $4$ | $30$ | $( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19, 22,24,26,27,30)$ |
| $ 6, 6, 6, 6, 6 $ | $5$ | $6$ | $( 1, 3,11,13,21,23)( 2, 4,12,14,22,24)( 5,20,15,30,25, 9)( 6,19,16,29,26,10) ( 7,28,17, 8,27,18)$ |
| $ 12, 12, 6 $ | $5$ | $12$ | $( 1, 3,17,26,21,23, 7,16,11,13,27, 6)( 2, 4,18,25,22,24, 8,15,12,14,28, 5) ( 9,29,20,10,30,19)$ |
| $ 12, 12, 6 $ | $5$ | $12$ | $( 1, 3,29, 8,21,23,19,28,11,13,10,18)( 2, 4,30, 7,22,24,20,27,12,14, 9,17) ( 5,26,15, 6,25,16)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1, 4, 5, 7,10,11,14,15,17,19,21,24,25,27,29)( 2, 3, 6, 8, 9,12,13,16,18,20, 22,23,26,28,30)$ |
| $ 6, 6, 6, 6, 3, 3 $ | $5$ | $6$ | $( 1, 4,11,14,21,24)( 2, 3,12,13,22,23)( 5,19,15,29,25,10)( 6,20,16,30,26, 9) ( 7,27,17)( 8,28,18)$ |
| $ 12, 12, 3, 3 $ | $5$ | $12$ | $( 1, 4,17,25,21,24, 7,15,11,14,27, 5)( 2, 3,18,26,22,23, 8,16,12,13,28, 6) ( 9,30,20)(10,29,19)$ |
| $ 12, 12, 3, 3 $ | $5$ | $12$ | $( 1, 4,29, 7,21,24,19,27,11,14,10,17)( 2, 3,30, 8,22,23,20,28,12,13, 9,18) ( 5,25,15)( 6,26,16)$ |
| $ 12, 12, 3, 3 $ | $5$ | $12$ | $( 1, 5,27,14,11,15, 7,24,21,25,17, 4)( 2, 6,28,13,12,16, 8,23,22,26,18, 3) ( 9,20,30)(10,19,29)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1, 5,10,14,17,21,25,29, 4, 7,11,15,19,24,27)( 2, 6, 9,13,18,22,26,30, 3, 8, 12,16,20,23,28)$ |
| $ 6, 6, 6, 6, 3, 3 $ | $5$ | $6$ | $( 1, 5,21,25,11,15)( 2, 6,22,26,12,16)( 3,13,23)( 4,14,24)( 7,29,27,19,17,10) ( 8,30,28,20,18, 9)$ |
| $ 12, 12, 3, 3 $ | $5$ | $12$ | $( 1, 5, 4,19,11,15,14,29,21,25,24,10)( 2, 6, 3,20,12,16,13,30,22,26,23, 9) ( 7,17,27)( 8,18,28)$ |
| $ 12, 12, 6 $ | $5$ | $12$ | $( 1, 6,27,13,11,16, 7,23,21,26,17, 3)( 2, 5,28,14,12,15, 8,24,22,25,18, 4) ( 9,19,30,10,20,29)$ |
| $ 30 $ | $4$ | $30$ | $( 1, 6,10,13,17,22,25,30, 4, 8,11,16,19,23,27, 2, 5, 9,14,18,21,26,29, 3, 7, 12,15,20,24,28)$ |
| $ 6, 6, 6, 6, 6 $ | $5$ | $6$ | $( 1, 6,21,26,11,16)( 2, 5,22,25,12,15)( 3,14,23, 4,13,24)( 7,30,27,20,17, 9) ( 8,29,28,19,18,10)$ |
| $ 12, 12, 6 $ | $5$ | $12$ | $( 1, 6, 4,20,11,16,14,30,21,26,24, 9)( 2, 5, 3,19,12,15,13,29,22,25,23,10) ( 7,18,27, 8,17,28)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$ |
| $ 10, 10, 10 $ | $4$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,11,21)( 2,12,22)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,20,30)(10,19,29)$ |
| $ 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,21,11)( 2,22,12)( 3,23,13)( 4,24,14)( 5,25,15)( 6,26,16)( 7,27,17) ( 8,28,18)( 9,30,20)(10,29,19)$ |
| $ 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,22,11, 2,21,12)( 3,24,13, 4,23,14)( 5,26,15, 6,25,16)( 7,28,17, 8,27,18) ( 9,29,20,10,30,19)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $120=2^{3} \cdot 3 \cdot 5$ | 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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| Label: | 120.40 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);