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Magma
magma: G := TransitiveGroup(45, 23);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_9\times F_5$ | ||
| 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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| $\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,36,40,37,17,5,10,8,31,20,25,24,3,35,42,39,18,6,12,9,33,21,26,23,2,34,41,38,16,4,11,7,32,19,27,22)(13,30,45,15,29,43,14,28,44), (1,4,17,19,31,36,3,5,18,20,33,35,2,6,16,21,32,34)(7,28,22,45,37,14,8,30,24,43,39,13,9,29,23,44,38,15)(10,41,25,11,42,27,12,40,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $9$: $C_9$ $12$: $C_{12}$ $18$: $C_{18}$ $20$: $F_5$ $36$: $C_{36}$ $60$: $F_5\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $F_5$
Degree 9: $C_9$
Degree 15: $F_5\times C_3$
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 $ | $5$ | $2$ | $( 4,15)( 5,14)( 6,13)( 7,26)( 8,27)( 9,25)(10,39)(11,37)(12,38)(19,28)(20,30) (21,29)(22,41)(23,42)(24,40)(34,44)(35,43)(36,45)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 4,24,15,40)( 5,23,14,42)( 6,22,13,41)( 7,43,26,35)( 8,44,27,34)( 9,45,25,36) (10,19,39,28)(11,21,37,29)(12,20,38,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 4,40,15,24)( 5,42,14,23)( 6,41,13,22)( 7,35,26,43)( 8,34,27,44)( 9,36,25,45) (10,28,39,19)(11,29,37,21)(12,30,38,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 2, 3)( 4,13, 5,15, 6,14)( 7,25, 8,26, 9,27)(10,37,12,39,11,38)(16,18,17) (19,29,20,28,21,30)(22,42,24,41,23,40)(31,32,33)(34,43,36,44,35,45)$ |
| $ 12, 12, 12, 3, 3, 3 $ | $5$ | $12$ | $( 1, 2, 3)( 4,22,14,40, 6,23,15,41, 5,24,13,42)( 7,45,27,35, 9,44,26,36, 8,43, 25,34)(10,21,38,28,11,20,39,29,12,19,37,30)(16,18,17)(31,32,33)$ |
| $ 12, 12, 12, 3, 3, 3 $ | $5$ | $12$ | $( 1, 2, 3)( 4,41,14,24, 6,42,15,22, 5,40,13,23)( 7,36,27,43, 9,34,26,45, 8,35, 25,44)(10,29,38,19,11,30,39,21,12,28,37,20)(16,18,17)(31,32,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,20,21) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41) (43,44,45)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 3, 2)( 4,14, 6,15, 5,13)( 7,27, 9,26, 8,25)(10,38,11,39,12,37)(16,17,18) (19,30,21,28,20,29)(22,40,23,41,24,42)(31,33,32)(34,45,35,44,36,43)$ |
| $ 12, 12, 12, 3, 3, 3 $ | $5$ | $12$ | $( 1, 3, 2)( 4,23,13,40, 5,22,15,42, 6,24,14,41)( 7,44,25,35, 8,45,26,34, 9,43, 27,36)(10,20,37,28,12,21,39,30,11,19,38,29)(16,17,18)(31,33,32)$ |
| $ 12, 12, 12, 3, 3, 3 $ | $5$ | $12$ | $( 1, 3, 2)( 4,42,13,24, 5,41,15,23, 6,40,14,22)( 7,34,25,43, 8,36,26,44, 9,35, 27,45)(10,30,37,19,12,29,39,20,11,28,38,21)(16,17,18)(31,33,32)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 4, 7,10,13,16,21,23,27,30,33,35,39,41,45, 3, 5, 8,12,15,17,19,22,25,29, 32,34,38,40,43, 2, 6, 9,11,14,18,20,24,26,28,31,36,37,42,44)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 4,17,19,31,36, 3, 5,18,20,33,35, 2, 6,16,21,32,34)( 7,28,22,45,37,14, 8, 30,24,43,39,13, 9,29,23,44,38,15)(10,41,25,11,42,27,12,40,26)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 4,26,39,31,36,11,23,18,20,40, 7, 2, 6,25,37,32,34,12,24,17,19,41, 9, 3, 5,27,38,33,35,10,22,16,21,42, 8)(13,45,29,14,44,30,15,43,28)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 4,43,10,31,36,29,42,18,20,15,26, 2, 6,45,11,32,34,30,40,17,19,13,25, 3, 5,44,12,33,35,28,41,16,21,14,27)( 7,39,22, 9,37,23, 8,38,24)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 5, 9,10,15,18,21,22,26,30,32,36,39,40,44, 3, 6, 7,12,14,16,19,24,27,29, 31,35,38,42,45, 2, 4, 8,11,13,17,20,23,25,28,33,34,37,41,43)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 5,16,19,33,34, 3, 6,17,20,32,36, 2, 4,18,21,31,35)( 7,30,23,45,39,15, 8, 29,22,43,38,14, 9,28,24,44,37,13)(10,40,27,11,41,26,12,42,25)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 5,25,39,33,34,11,22,17,20,42, 9, 2, 4,27,37,31,35,12,23,16,19,40, 8, 3, 6,26,38,32,36,10,24,18,21,41, 7)(13,43,30,14,45,28,15,44,29)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 5,45,10,33,34,29,41,17,20,14,25, 2, 4,44,11,31,35,30,42,16,19,15,27, 3, 6,43,12,32,36,28,40,18,21,13,26)( 7,38,23, 9,39,24, 8,37,22)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 6, 8,10,14,17,21,24,25,30,31,34,39,42,43, 3, 4, 9,12,13,18,19,23,26,29, 33,36,38,41,44, 2, 5, 7,11,15,16,20,22,27,28,32,35,37,40,45)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 6,18,19,32,35, 3, 4,16,20,31,34, 2, 5,17,21,33,36)( 7,29,24,45,38,13, 8, 28,23,43,37,15, 9,30,22,44,39,14)(10,42,26,11,40,25,12,41,27)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 6,27,39,32,35,11,24,16,20,41, 8, 2, 5,26,37,33,36,12,22,18,19,42, 7, 3, 4,25,38,31,34,10,23,17,21,40, 9)(13,44,28,14,43,29,15,45,30)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 6,44,10,32,35,29,40,16,20,13,27, 2, 5,43,11,33,36,30,41,18,19,14,26, 3, 4,45,12,31,34,28,42,17,21,15,25)( 7,37,24, 9,38,22, 8,39,23)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 7,41,21,18,24,10,36,32,38,26, 6, 3, 8,40,19,16,23,12,35,31,37,27, 4, 2, 9,42,20,17,22,11,34,33,39,25, 5)(13,29,44,15,28,45,14,30,43)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 7,13,21,27,33,39,45, 5,12,17,22,29,34,40, 2, 9,14,20,26,31,37,44, 4,10, 16,23,30,35,41, 3, 8,15,19,25,32,38,43, 6,11,18,24,28,36,42)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 7,31,37,18,24, 2, 9,32,38,17,22, 3, 8,33,39,16,23)( 4,19,36, 5,20,35, 6, 21,34)(10,45,42,30,26,13,11,44,40,28,25,14,12,43,41,29,27,15)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 7, 6,29,18,24,19,45,32,38,35,13, 3, 8, 4,28,16,23,20,43,31,37,34,15, 2, 9, 5,30,17,22,21,44,33,39,36,14)(10,25,42,12,26,41,11,27,40)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 8,42,21,16,22,10,35,33,38,27, 5, 3, 9,41,19,17,24,12,34,32,37,25, 6, 2, 7,40,20,18,23,11,36,31,39,26, 4)(13,28,43,15,30,44,14,29,45)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 8,14,21,25,31,39,43, 4,12,18,23,29,36,41, 2, 7,15,20,27,32,37,45, 6,10, 17,24,30,34,42, 3, 9,13,19,26,33,38,44, 5,11,16,22,28,35,40)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 8,32,37,16,22, 2, 7,33,38,18,23, 3, 9,31,39,17,24)( 4,20,34, 5,21,36, 6, 19,35)(10,43,40,30,27,14,11,45,41,28,26,15,12,44,42,29,25,13)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 8, 5,29,16,22,19,43,33,38,34,14, 3, 9, 6,28,17,24,20,44,32,37,36,13, 2, 7, 4,30,18,23,21,45,31,39,35,15)(10,26,40,12,27,42,11,25,41)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 9,40,21,17,23,10,34,31,38,25, 4, 3, 7,42,19,18,22,12,36,33,37,26, 5, 2, 8,41,20,16,24,11,35,32,39,27, 6)(13,30,45,15,29,43,14,28,44)$ |
| $ 45 $ | $4$ | $45$ | $( 1, 9,15,21,26,32,39,44, 6,12,16,24,29,35,42, 2, 8,13,20,25,33,37,43, 5,10, 18,22,30,36,40, 3, 7,14,19,27,31,38,45, 4,11,17,23,28,34,41)$ |
| $ 18, 18, 9 $ | $5$ | $18$ | $( 1, 9,33,37,17,23, 2, 8,31,38,16,24, 3, 7,32,39,18,22)( 4,21,35, 5,19,34, 6, 20,36)(10,44,41,30,25,15,11,43,42,28,27,13,12,45,40,29,26,14)$ |
| $ 36, 9 $ | $5$ | $36$ | $( 1, 9, 4,29,17,23,19,44,31,38,36,15, 3, 7, 5,28,18,22,20,45,33,37,35,14, 2, 8, 6,30,16,24,21,43,32,39,34,13)(10,27,41,12,25,40,11,26,42)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,10,21,30,39, 3,12,19,29,38, 2,11,20,28,37)( 4,13,23,33,41, 5,15,22,32,40, 6,14,24,31,42)( 7,16,27,35,45, 8,17,25,34,43, 9,18,26,36,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,11,19,30,37, 2,12,21,28,38, 3,10,20,29,39)( 4,14,22,33,42, 6,15,23,31,40, 5,13,24,32,41)( 7,18,25,35,44, 9,17,27,36,43, 8,16,26,34,45)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,12,20,30,38)( 2,10,19,28,39)( 3,11,21,29,37)( 4,15,24,33,40) ( 5,14,23,32,42)( 6,13,22,31,41)( 7,17,26,35,43)( 8,18,27,34,44) ( 9,16,25,36,45)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,16,33, 3,17,32, 2,18,31)( 4,21,35, 5,19,34, 6,20,36)( 7,23,39, 8,22,38, 9, 24,37)(10,27,41,12,25,40,11,26,42)(13,30,45,15,29,43,14,28,44)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,17,31, 3,18,33, 2,16,32)( 4,19,36, 5,20,35, 6,21,34)( 7,22,37, 8,24,39, 9, 23,38)(10,25,42,12,26,41,11,27,40)(13,29,44,15,28,45,14,30,43)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,18,32, 3,16,31, 2,17,33)( 4,20,34, 5,21,36, 6,19,35)( 7,24,38, 8,23,37, 9, 22,39)(10,26,40,12,27,42,11,25,41)(13,28,43,15,30,44,14,29,45)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,31,18, 2,32,17, 3,33,16)( 4,36,20, 6,34,19, 5,35,21)( 7,37,24, 9,38,22, 8, 39,23)(10,42,26,11,40,25,12,41,27)(13,44,28,14,43,29,15,45,30)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,32,16, 2,33,18, 3,31,17)( 4,34,21, 6,35,20, 5,36,19)( 7,38,23, 9,39,24, 8, 37,22)(10,40,27,11,41,26,12,42,25)(13,43,30,14,45,28,15,44,29)$ |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,33,17, 2,31,16, 3,32,18)( 4,35,19, 6,36,21, 5,34,20)( 7,39,22, 9,37,23, 8, 38,24)(10,41,25,11,42,27,12,40,26)(13,45,29,14,44,30,15,43,28)$ |
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: | 180.5 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);