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Magma
magma: G := TransitiveGroup(45, 17);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{45}:C_4$ | ||
| 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,26,32,12,17,40,2,25,31,10,16,41,3,27,33,11,18,42)(4,37,36,24,21,7,6,39,35,23,20,8,5,38,34,22,19,9)(13,28,45,14,29,43,15,30,44), (1,31)(2,32)(3,33)(4,11,13,38)(5,12,14,37)(6,10,15,39)(7,35,25,44)(8,36,27,45)(9,34,26,43)(16,18)(19,42,30,22)(20,40,28,24)(21,41,29,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ $12$: $C_3 : C_4$ $18$: $D_{9}$ $20$: $F_5$ $36$: 36T9 $60$: $C_{15} : C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: $F_5$
Degree 9: $D_{9}$
Degree 15: $C_{15} : C_4$
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,25)( 8,27)( 9,26)(10,39)(11,38)(12,37)(19,30)(20,28) (21,29)(22,42)(23,41)(24,40)(34,43)(35,44)(36,45)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ | $45$ | $4$ | $( 2, 3)( 4, 7,13,25)( 5, 8,14,27)( 6, 9,15,26)(10,19,39,30)(11,20,38,28) (12,21,37,29)(16,31)(17,33)(18,32)(22,44,42,35)(23,43,41,34)(24,45,40,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ | $45$ | $4$ | $( 2, 3)( 4,25,13, 7)( 5,27,14, 8)( 6,26,15, 9)(10,30,39,19)(11,28,38,20) (12,29,37,21)(16,31)(17,33)(18,32)(22,35,42,44)(23,34,41,43)(24,36,40,45)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 2, 3)( 4,15, 5,13, 6,14)( 7,27, 9,25, 8,26)(10,38,12,39,11,37)(16,18,17) (19,29,20,30,21,28)(22,40,23,42,24,41)(31,33,32)(34,45,35,43,36,44)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 4, 8,11,15,16,20,24,25,30,32,36,38,42,44, 3, 5, 7,10,13,17,21,22,26,28, 33,34,39,41,45, 2, 6, 9,12,14,18,19,23,27,29,31,35,37,40,43)$ |
| $ 18, 18, 9 $ | $10$ | $18$ | $( 1, 4,17,21,31,35, 3, 5,18,19,32,36, 2, 6,16,20,33,34)( 7,30,23,45,38,15, 9, 28,24,43,39,13, 8,29,22,44,37,14)(10,41,27,11,42,26,12,40,25)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 5, 9,11,13,18,20,22,27,30,33,35,38,41,43, 3, 6, 8,10,14,16,21,23,25,28, 31,36,39,40,44, 2, 4, 7,12,15,17,19,24,26,29,32,34,37,42,45)$ |
| $ 18, 18, 9 $ | $10$ | $18$ | $( 1, 5,16,21,32,34, 3, 6,17,19,33,35, 2, 4,18,20,31,36)( 7,28,22,45,39,14, 9, 29,23,43,37,15, 8,30,24,44,38,13)(10,40,26,11,41,25,12,42,27)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 6, 7,11,14,17,20,23,26,30,31,34,38,40,45, 3, 4, 9,10,15,18,21,24,27,28, 32,35,39,42,43, 2, 5, 8,12,13,16,19,22,25,29,33,36,37,41,44)$ |
| $ 18, 18, 9 $ | $10$ | $18$ | $( 1, 6,18,21,33,36, 3, 4,16,19,31,34, 2, 5,17,20,32,35)( 7,29,24,45,37,13, 9, 30,22,43,38,14, 8,28,23,44,39,15)(10,42,25,11,40,27,12,41,26)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,19,30,39)( 2,11,21,29,38)( 3,12,20,28,37)( 4,13,23,32,41) ( 5,14,24,33,40)( 6,15,22,31,42)( 7,18,25,34,43)( 8,17,27,36,45) ( 9,16,26,35,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,11,20,30,38, 3,10,21,28,39, 2,12,19,29,37)( 4,15,24,32,42, 5,13,22,33,41, 6,14,23,31,40)( 7,17,26,34,45, 9,18,27,35,43, 8,16,25,36,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,12,21,30,37, 2,10,20,29,39, 3,11,19,28,38)( 4,14,22,32,40, 6,13,24,31,41, 5,15,23,33,42)( 7,16,27,34,44, 8,18,26,36,43, 9,17,25,35,45)$ |
| $ 9, 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,16,32, 3,17,33, 2,18,31)( 4,20,36, 5,21,34, 6,19,35)( 7,22,39, 9,23,37, 8, 24,38)(10,26,41,12,27,40,11,25,42)(13,28,45,14,29,43,15,30,44)$ |
| $ 9, 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,17,31, 3,18,32, 2,16,33)( 4,21,35, 5,19,36, 6,20,34)( 7,23,38, 9,24,39, 8, 22,37)(10,27,42,12,25,41,11,26,40)(13,29,44,14,30,45,15,28,43)$ |
| $ 9, 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,18,33, 3,16,31, 2,17,32)( 4,19,34, 5,20,35, 6,21,36)( 7,24,37, 9,22,38, 8, 23,39)(10,25,40,12,26,42,11,27,41)(13,30,43,14,28,44,15,29,45)$ |
| $ 45 $ | $4$ | $45$ | $( 1,22,43,21,40,17,37,13,35,10,31, 7,29, 5,27, 3,23,44,19,42,18,38,14,36,12, 32, 9,30, 6,25, 2,24,45,20,41,16,39,15,34,11,33, 8,28, 4,26)$ |
| $ 45 $ | $4$ | $45$ | $( 1,23,45,21,42,16,37,14,34,10,32, 8,29, 6,26, 3,24,43,19,41,17,38,15,35,12, 33, 7,30, 4,27, 2,22,44,20,40,18,39,13,36,11,31, 9,28, 5,25)$ |
| $ 45 $ | $4$ | $45$ | $( 1,24,44,21,41,18,37,15,36,10,33, 9,29, 4,25, 3,22,45,19,40,16,38,13,34,12, 31, 8,30, 5,26, 2,23,43,20,42,17,39,14,35,11,32, 7,28, 6,27)$ |
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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| Label: | 180.6 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);