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Magma
magma: G := TransitiveGroup(45, 21);
Group action invariants
| Degree $n$: | $45$ | 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: | $C_{15}:D_6$ | ||
| 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: | (1,7,29,35,10,17,37,44,21,25)(2,9,30,36,11,18,39,45,19,26)(3,8,28,34,12,16,38,43,20,27)(4,42,31,22,15,5,41,32,23,13)(6,40,33,24,14), (1,28,2,29,3,30)(4,25,6,27,5,26)(7,24,8,22,9,23)(10,20,11,21,12,19)(13,18,15,17,14,16)(31,44,33,43,32,45)(34,42,36,41,35,40)(37,38,39) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $10$: $D_{5}$ $12$: $D_{6}$ x 2 $20$: $D_{10}$ $36$: $S_3^2$ $60$: $D_5\times S_3$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 2
Degree 5: $D_{5}$
Degree 9: $S_3^2$
Degree 15: $D_5\times S_3$ x 2
Low degree siblings
30T43Siblings 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, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $15$ | $2$ | $( 4,43)( 5,44)( 6,45)( 7,42)( 8,41)( 9,40)(10,37)(11,39)(12,38)(13,35)(14,36) (15,34)(16,31)(17,32)(18,33)(19,30)(20,28)(21,29)(22,25)(23,27)(24,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $15$ | $2$ | $( 2, 3)( 4,15)( 5,14)( 6,13)( 7,26)( 8,27)( 9,25)(10,37)(11,38)(12,39)(17,18) (19,28)(20,30)(21,29)(22,40)(23,41)(24,42)(32,33)(34,43)(35,45)(36,44)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 4,34)( 5,36)( 6,35)( 7,24)( 8,23)( 9,22)(11,12)(13,45)(14,44)(15,43) (16,31)(17,33)(18,32)(19,20)(25,40)(26,42)(27,41)(28,30)(38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 3 $ | $30$ | $6$ | $( 1, 2, 3)( 4,45, 5,43, 6,44)( 7,41, 9,42, 8,40)(10,39,12,37,11,38) (13,34,14,35,15,36)(16,33,17,31,18,32)(19,28,21,30,20,29)(22,27,24,25,23,26)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 3 $ | $30$ | $6$ | $( 1, 4,16,21,31,34)( 2, 5,18,20,33,35)( 3, 6,17,19,32,36)( 7,30,22,45,38,14) ( 8,29,23,43,37,15)( 9,28,24,44,39,13)(10,41,27)(11,42,26,12,40,25)$ |
| $ 10, 10, 10, 10, 5 $ | $18$ | $10$ | $( 1, 4,37,41,29,31,21,23,10,15)( 2, 5,39,42,30,32,19,22,11,13)( 3, 6,38,40,28, 33,20,24,12,14)( 7,26,44,18,35, 9,25,45,17,36)( 8,27,43,16,34)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 4, 8,10,15,16,21,23,27,29,31,34,37,41,43)( 2, 6, 9,11,14,18,19,24,26,30, 33,36,39,40,45)( 3, 5, 7,12,13,17,20,22,25,28,32,35,38,42,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 5, 9,10,13,18,21,22,26,29,32,36,37,42,45)( 2, 4, 7,11,15,17,19,23,25,30, 31,35,39,41,44)( 3, 6, 8,12,14,16,20,24,27,28,33,34,38,40,43)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 6, 7,10,14,17,21,24,25,29,33,35,37,40,44)( 2, 5, 8,11,13,16,19,22,27,30, 32,34,39,42,43)( 3, 4, 9,12,15,18,20,23,26,28,31,36,38,41,45)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 7,14,21,25,33,37,44, 6,10,17,24,29,35,40)( 2, 8,13,19,27,32,39,43, 5,11, 16,22,30,34,42)( 3, 9,15,20,26,31,38,45, 4,12,18,23,28,36,41)$ |
| $ 10, 10, 10, 10, 5 $ | $18$ | $10$ | $( 1, 7,29,35,10,17,37,44,21,25)( 2, 9,30,36,11,18,39,45,19,26)( 3, 8,28,34,12, 16,38,43,20,27)( 4,42,31,22,15, 5,41,32,23,13)( 6,40,33,24,14)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 8,15,21,27,31,37,43, 4,10,16,23,29,34,41)( 2, 9,14,19,26,33,39,45, 6,11, 18,24,30,36,40)( 3, 7,13,20,25,32,38,44, 5,12,17,22,28,35,42)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 9,13,21,26,32,37,45, 5,10,18,22,29,36,42)( 2, 7,15,19,25,31,39,44, 4,11, 17,23,30,35,41)( 3, 8,14,20,27,33,38,43, 6,12,16,24,28,34,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,21,29,37)( 2,11,19,30,39)( 3,12,20,28,38)( 4,15,23,31,41) ( 5,13,22,32,42)( 6,14,24,33,40)( 7,17,25,35,44)( 8,16,27,34,43) ( 9,18,26,36,45)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,11,20,29,39, 3,10,19,28,37, 2,12,21,30,38)( 4,14,22,31,40, 5,15,24,32,41, 6,13,23,33,42)( 7,16,26,35,43, 9,17,27,36,44, 8,18,25,34,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,31)( 2,18,33)( 3,17,32)( 4,21,34)( 5,20,35)( 6,19,36)( 7,22,38) ( 8,23,37)( 9,24,39)(10,27,41)(11,26,40)(12,25,42)(13,28,44)(14,30,45) (15,29,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,17,33)( 2,16,32)( 3,18,31)( 4,20,36)( 5,19,34)( 6,21,35)( 7,24,37) ( 8,22,39)( 9,23,38)(10,25,40)(11,27,42)(12,26,41)(13,30,43)(14,29,44) (15,28,45)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,19,38,10,30, 3,21,39,12,29, 2,20,37,11,28)( 4,24,42,15,33, 5,23,40,13,31, 6,22,41,14,32)( 7,27,45,17,34, 9,25,43,18,35, 8,26,44,16,36)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,21,37,10,29)( 2,19,39,11,30)( 3,20,38,12,28)( 4,23,41,15,31) ( 5,22,42,13,32)( 6,24,40,14,33)( 7,25,44,17,35)( 8,27,43,16,34) ( 9,26,45,18,36)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $180=2^{2} \cdot 3^{2} \cdot 5$ | 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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| Nilpotency class: | not nilpotent | ||
| Label: | 180.30 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);