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Magma
magma: G := TransitiveGroup(45, 5);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times D_{15}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,26,4,30,9,32,11,35,13,37,17,40,20,43,22)(2,25,6,29,8,33,12,36,14,39,16,41,21,45,23)(3,27,5,28,7,31,10,34,15,38,18,42,19,44,24), (1,21,3,20,2,19)(4,16,5,17,6,18)(7,13,8,15,9,14)(10,11,12)(22,45,24,43,23,44)(25,42,26,41,27,40)(28,37,29,38,30,39)(31,35,33,34,32,36) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $10$: $D_{5}$ $18$: $S_3\times C_3$ $30$: $D_{15}$, $D_5\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 9: $S_3\times C_3$
Degree 15: $D_{15}$, $D_5\times C_3$
Low degree siblings
30T16Siblings 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 $ | $15$ | $2$ | $( 4,43)( 5,44)( 6,45)( 7,42)( 8,41)( 9,40)(10,38)(11,37)(12,39)(13,35)(14,36) (15,34)(16,33)(17,32)(18,31)(19,28)(20,30)(21,29)(22,26)(23,25)(24,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,35,36)(37,39,38)(40,41,42) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 3 $ | $15$ | $6$ | $( 1, 2, 3)( 4,45, 5,43, 6,44)( 7,40, 8,42, 9,41)(10,37,12,38,11,39) (13,36,15,35,14,34)(16,31,17,33,18,32)(19,30,21,28,20,29)(22,25,24,26,23,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,24,23)(25,26,27)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,44,45)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 3 $ | $15$ | $6$ | $( 1, 3, 2)( 4,44, 6,43, 5,45)( 7,41, 9,42, 8,40)(10,39,11,38,12,37) (13,34,14,35,15,36)(16,32,18,33,17,31)(19,29,20,28,21,30)(22,27,23,26,24,25)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 4, 9,11,13,17,20,22,26,30,32,35,37,40,43)( 2, 6, 8,12,14,16,21,23,25,29, 33,36,39,41,45)( 3, 5, 7,10,15,18,19,24,27,28,31,34,38,42,44)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 5, 8,11,15,16,20,24,25,30,31,36,37,42,45)( 2, 4, 7,12,13,18,21,22,27,29, 32,34,39,40,44)( 3, 6, 9,10,14,17,19,23,26,28,33,35,38,41,43)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 6, 7,11,14,18,20,23,27,30,33,34,37,41,44)( 2, 5, 9,12,15,17,21,24,26,29, 31,35,39,42,43)( 3, 4, 8,10,13,16,19,22,25,28,32,36,38,40,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 7,14,20,27,33,37,44, 6,11,18,23,30,34,41)( 2, 9,15,21,26,31,39,43, 5,12, 17,24,29,35,42)( 3, 8,13,19,25,32,38,45, 4,10,16,22,28,36,40)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 8,15,20,25,31,37,45, 5,11,16,24,30,36,42)( 2, 7,13,21,27,32,39,44, 4,12, 18,22,29,34,40)( 3, 9,14,19,26,33,38,43, 6,10,17,23,28,35,41)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1, 9,13,20,26,32,37,43, 4,11,17,22,30,35,40)( 2, 8,14,21,25,33,39,45, 6,12, 16,23,29,36,41)( 3, 7,15,19,27,31,38,44, 5,10,18,24,28,34,42)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,10,21,30,38, 2,11,19,29,37, 3,12,20,28,39)( 4,15,23,32,42, 6,13,24,33,40, 5,14,22,31,41)( 7,16,26,34,45, 9,18,25,35,44, 8,17,27,36,43)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,20,30,37)( 2,12,21,29,39)( 3,10,19,28,38)( 4,13,22,32,40) ( 5,15,24,31,42)( 6,14,23,33,41)( 7,18,27,34,44)( 8,16,25,36,45) ( 9,17,26,35,43)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,12,19,30,39, 3,11,21,28,37, 2,10,20,29,38)( 4,14,24,32,41, 5,13,23,31,40, 6,15,22,33,42)( 7,17,25,34,43, 8,18,26,36,44, 9,16,27,35,45)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,13,26,37, 4,17,30,40, 9,20,32,43,11,22,35)( 2,14,25,39, 6,16,29,41, 8,21, 33,45,12,23,36)( 3,15,27,38, 5,18,28,42, 7,19,31,44,10,24,34)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,14,27,37, 6,18,30,41, 7,20,33,44,11,23,34)( 2,15,26,39, 5,17,29,42, 9,21, 31,43,12,24,35)( 3,13,25,38, 4,16,28,40, 8,19,32,45,10,22,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,15,25,37, 5,16,30,42, 8,20,31,45,11,24,36)( 2,13,27,39, 4,18,29,40, 7,21, 32,44,12,22,34)( 3,14,26,38, 6,17,28,41, 9,19,33,43,10,23,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,31)( 2,18,32)( 3,17,33)( 4,21,34)( 5,20,36)( 6,19,35)( 7,22,39) ( 8,24,37)( 9,23,38)(10,26,41)(11,25,42)(12,27,40)(13,29,44)(14,28,43) (15,30,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,32)( 2,16,33)( 3,18,31)( 4,20,35)( 5,19,34)( 6,21,36)( 7,24,38) ( 8,23,39)( 9,22,37)(10,27,42)(11,26,40)(12,25,41)(13,30,43)(14,29,45) (15,28,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,18,33)( 2,17,31)( 3,16,32)( 4,19,36)( 5,21,35)( 6,20,34)( 7,23,37) ( 8,22,38)( 9,24,39)(10,25,40)(11,27,41)(12,26,42)(13,28,45)(14,30,44) (15,29,43)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,19,39,11,28, 2,20,38,12,30, 3,21,37,10,29)( 4,24,41,13,31, 6,22,42,14,32, 5,23,40,15,33)( 7,25,43,18,36, 9,27,45,17,34, 8,26,44,16,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,20,37,11,30)( 2,21,39,12,29)( 3,19,38,10,28)( 4,22,40,13,32) ( 5,24,42,15,31)( 6,23,41,14,33)( 7,27,44,18,34)( 8,25,45,16,36) ( 9,26,43,17,35)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,21,38,11,29, 3,20,39,10,30, 2,19,37,12,28)( 4,23,42,13,33, 5,22,41,15,32, 6,24,40,14,31)( 7,26,45,18,35, 8,27,43,16,34, 9,25,44,17,36)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,22,43,20,40,17,37,13,35,11,32, 9,30, 4,26)( 2,23,45,21,41,16,39,14,36,12, 33, 8,29, 6,25)( 3,24,44,19,42,18,38,15,34,10,31, 7,28, 5,27)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,23,44,20,41,18,37,14,34,11,33, 7,30, 6,27)( 2,24,43,21,42,17,39,15,35,12, 31, 9,29, 5,26)( 3,22,45,19,40,16,38,13,36,10,32, 8,28, 4,25)$ |
| $ 15, 15, 15 $ | $2$ | $15$ | $( 1,24,45,20,42,16,37,15,36,11,31, 8,30, 5,25)( 2,22,44,21,40,18,39,13,34,12, 32, 7,29, 4,27)( 3,23,43,19,41,17,38,14,35,10,33, 9,28, 6,26)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $90=2 \cdot 3^{2} \cdot 5$ | 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: | 90.7 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);